Direct and inverse scattering problems for the first-order discrete system associated with the derivative NLS system
aa r X i v : . [ m a t h - ph ] J a n Direct and inverse scattering problems for thefirst-order discrete system associated with thederivative NLS system
T. Aktosun and R. ErcanDepartment of MathematicsUniversity of Texas at ArlingtonArlington, TX 76019-0408, USA
Abstract
The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semi-discrete version of thederivative NLS system. The Jost solutions, the scattering coefficients,the bound-state dependency and norming constants are investigated andrelated to the corresponding quantities for two particular discrete linearsystems associated with the semi-discrete version of the NLS system. Thebound-state data set with any multiplicities is described in an elegantmanner in terms of a pair of constant matrix triplets. Several methodsare presented to the solve the inverse problem. One of these methods in-volves a discrete Marchenko system using as input the scattering data setconsisting of the scattering coefficients and the bound-state information,and this method is presented in a way generalizable to other first-ordersystems both in the discrete and continuous cases for which a Marchenkosystem is not yet available. Finally, using the time-evolved scatteringdata set, the inverse scattering transform is applied on the correspondingsemi-discrete derivative NLS system, and in the reflectionless case certainexplicit solution formulas are presented in closed form expressed in termsof the two matrix triplets.
In this paper we are interested in analyzing the direct and inverse scatteringproblems for the first-order discrete system " α n β n = z (cid:18) z − z (cid:19) q n z r n z + (cid:18) z − z (cid:19) q n r n " α n +1 β n +1 , n ∈ Z , (1.1)1here z is the spectral parameter taking values on the unit circle T in the com-plex z -plane C , n is the discrete independent variable taking values in the setof integers Z , the complex-valued scalar quantities q n and r n correspond therespective values evaluated at n for the potential pair ( q, r ) , and (cid:20) α n β n (cid:21) corre-sponds to the value of the wavefunction at the spacial location n. We assumethat q n and r n are rapidly decaying in the sense that they vanish faster thanany negative powers of | n | as n → ±∞ . We also assume that1 − q n r n = 0 , q n r n +1 = 0 , n ∈ Z . (1.2)The complex-valued quantities α n and β n depend on the spectral parameter z, but in our notation we usually suppress that z -dependence.The system in (1.1) is used as a model for an infinite lattice where theparticle with an internal structure at the lattice point n experiences local forcesfrom the potential values q n and r n . Since we assume that q n and r n vanishsufficiently fast as n → ±∞ , a scattering scenario can be established for (1.1).The direct scattering problem for (1.1) is described as the determination ofthe scattering data set consisting of the scattering coefficients and bound-stateinformation when the potential pair ( q, r ) is known. The inverse scatteringproblem for (1.1) consists of the recovery of the potential pair ( q, r ) when thescattering data set is given. Since q n and r n vanish sufficiently fast as n → ±∞ ,it follows from (1.1) that any solution to (1.1) has the asymptotic behavior " α n β n = " a ± z − n [1 + o (1)] b ± z n [1 + o (1)] , n → ±∞ , (1.3)for some constants a ± and b ± that may depend on z but not on n . By choosingtwo of the four coefficients a + , a − , b + , b − appearing in (1.3) in a specific way, weobtain a particular solution to (1.1). Note that (1.1) has two linearly indepen-dent solutions, and its general solution can be expressed as a linear combinationof any two linearly independent solutions.The discrete system (1.1) is related to the integrable semi-discrete system i ˙ q n + q n +1 − q n +1 r n +1 − q n − q n r n − q n q n r n +1 + q n − q n − r n = 0 ,i ˙ r n − r n +1 q n r n +1 + r n q n − r n + r n − q n r n − r n − − q n − r n − = 0 , (1.4)which is known as the semi-discrete derivative NLS (nonlinear Schr¨odinger)system or the semi-discrete Kaup-Newell system [11, 12, 13]. From the denom-inators in (1.4) we see why we need the restriction (1.2). Note that an overdotin (1.4) denotes the derivative with respect to the independent variable t, whichis interpreted as the time variable and is suppressed in (1.4). In our analysisof (1.1), without loss of generality we can either assume that q n and r n areindependent of t or they contain t as a parameter.2e analyze the direct and inverse scattering problems for (1.1) by using theconnection to the two first-order discrete systems " ξ n η n = z z u n z v n z " ξ n +1 η n +1 , n ∈ Z , (1.5) " γ n ǫ n = z z p n z s n z " γ n +1 ǫ n +1 , n ∈ Z , (1.6)where u n and v n are the values for the potential pair ( u, v ) and p n and s n arethe values for ( p, s ). By choosing ( u, v ) and ( p, s ) as in (3.1)–(3.4), we relate therelevant quantities for (1.1), (1.5), (1.6) to each other. Such relevant quantitiesinclude the Jost solutions, the scattering coefficients, and the bound-state datasets for each of (1.1), (1.5), (1.6).We remark that in the literature it is always assumed that the bound statesfor (1.1), (1.5), (1.6) are simple. In our paper we do not make such an arti-ficial assumption because we easily and in an elegant way handle the boundstates of any multiplicities, and this is done by using a pair of constant matrixtriplets describing the bound-state values of the spectral parameter z and thecorresponding norming constants.The systems (1.5) and (1.6) are of importance also in their own, and theyare known as the Ablowitz-Ladik systems or as the discrete AKNS systems. Itis possible [13] to transform (1.5) and (1.6) into " ˜ ξ n +1 ˜ η n +1 = z u n v n z " ˜ ξ n ˜ η n , n ∈ Z , (1.7) " ˜ γ n +1 ˜ ǫ n +1 = z p n s n z " ˜ γ n ˜ ǫ n , n ∈ Z . (1.8)Note that (1.5) and (1.7) also differ from each other by the fact that the appear-ances of the wavefunction values evaluated at n and n + 1 are switched. Thesame remark also applies to (1.6) and (1.8).As already pointed out by Tsuchida [13], the analysis of the direct andinverse scattering problems for an Ablowitz-Ladik system written in the form of(1.7) and (1.8) is unnecessarily complicated. For example, the analysis providedin [2] for (1.7) involves separating the scattering data into two parts containingeven and odd integer powers of z, respectively. This unnecessarily makes theanalysis cumbersome. Furthermore, if we use (1.7) with the roles of n and n + 1switched compared to (1.5) and use the scattering coefficients from the rightinstead of the scattering coefficients from the left as input, then the analysisof the inverse scattering problem for (1.7) by the Marchenko method becomesunnecessarily complicated. 3he researchers who are mainly interested in nonlinear evolution equationsuse only the scattering coefficient from the right without referring to the scatter-ing coefficients from the left. In this paper, we are careful in making a distinctionbetween the right and left scattering data sets. The right and left transmissioncoefficients in a first-order discrete linear system are unequal unless the coef-ficient matrix in that system has determinant equal to 1 . One can verify thatthe coefficient matrix in (1.1) has its determinant equal to 1 , whereas the corre-sponding determinants for (1.5) and (1.6) are given by 1 − u n v n and 1 − p n s n ,respectively. Thus, the left and right transmission coefficients for each of (1.5)and (1.6) are unequal.The scattering and inverse scattering problems for (1.1) have partially beenanalyzed by Tsuchida in [13]. Our own analysis is complementary to Tsuchida’swork in the following sense. Tsuchida’s main interest in (1.1) is confined toits relation to (1.4), and he only deals with the right scattering coefficients.Tsuchida exploits certain gauge transformations to relate (1.1) to two discreteAblowitz-Ladik systems, and he assumes that the bound states are all simple.Tsuchida’s expressions for the scattering coefficients not only involve the Jostsolutions to the relevant linear system but also the Jost solutions to the cor-responding adjoint system, whereas in our case the scattering coefficients areexpressed in terms of the Jost solutions to the relevant linear system only. Inour opinion the latter description of the scattering coefficients provides physicalinsight and intuition into the analysis of direct and inverse problems. Tsuchidaformulates a Marchenko system given in (4.12c) and (4.12d) of [13], somehowsimilar to our own alternate Marchenko system (7.7) and (7.8), but it lacks theappropriate symmetries existing in our alternate Marchenko system. In for-mulating his Marchenko system Tsuchida uses a Fourier transformation withrespect to z and not with respect to z . Furthermore, in Tsuchida’s formulationit is not quite clear how the scattering data sets for (1.1), (1.5), (1.6) are relatedto each other.One of the important accomplishments of our paper is the introduction ofa standard Marchenko formalism for (1.1) using as input the scattering datafrom (1.1) only. The formulation of our standard Marchenko system (6.1) is asignificant generalization step to solve inverse problems for various other dis-crete and continuous systems for which a standard Marchenko theory has notyet been formulated. As mentioned already, we also introduce an alternateMarchenko formalism for (1.1) using as input the scattering data sets from(1.5) and (1.6). Both our standard and alternate Marchenko systems we in-troduce have the appropriate symmetry properties and resemble the standardMarchenko systems arising in other continuous and discrete systems. The alter-nate Marchenko method in our paper corresponds to the discrete analog of thesystematic approach [5] we presented to solve the inverse scattering problem forthe energy-dependent AKNS system given in (1.1) of [5]. Besides [5] the mostrelevant reference for our current work is the important paper by Tsuchida [13].Our paper is organized as follows. In Section 2 we introduce the Jost solu-tions and the scattering coefficients for each of (1.1), (1.5), (1.6) and we presentsome relevant properties of those Jost solutions and scattering coefficients. In4hat section we also prove that the linear dependence of the appropriate pairs ofJost solutions occurs at the poles of the corresponding transmission coefficientsfor each of (1.1), (1.5), (1.6). In Section 3 when the corresponding potentialpairs are related to each other as in (3.1)–(3.4), we relate the Jost solutionsand scattering coefficients for (1.1) to those for (1.5) and (1.6). In that sectionwe also present certain relevant properties of the Jost solutions to (1.1) andexpress the potentials q n and r n in terms of the values at z = 1 of the Jostsolutions to (1.5) and (1.6). In Section 4 we describe the bound-state data setsfor each of (1.1), (1.5), (1.6) in terms of two matrix triplets, which allows usto handle bound states of any multiplicities in a systematic manner that canalso be used for other systems both in the continuous and discrete cases. Inthe formulation of the Marchenko method we show how the Marchenko kernelscontain the matrix triplets in a simple and elegant manner. Also in that section,when the potential pairs for (1.1), (1.5), (1.6) are related as in (3.1)–(3.4), weshow how the corresponding bound-state data sets are related to each other. InSection 5 we outline the steps to solve the direct problem for (1.1). In Section 6we introduce the Marchenko system (6.1) using as input the scattering datadirectly related to (1.1) and we describe how the potentials q n and r n are recov-ered from the solution (6.1). In Section 7 we present our alternate Marchenkosystem given in (7.7) and (7.8) using as input the scattering data sets from (1.5)and (1.6), as we also show how q n and r n are recovered from the solution tothe alternate Marchenko system. In Section 8 we describe various methods tosolve the inverse problem for (1.1) by using as input the scattering data for (1.1)and outline how the potentials q n and r n are recovered. Finally, in Section 9 weprovide the solution to the integrable nonlinear system (1.4) via the inverse scat-tering transform. This is done by providing the time evolution of the scatteringdata for (1.1) and by determining the corresponding time-evolved potentials q n and r n . In that section we also present some explicit solution formulas for (1.4)corresponding to time-evolved reflectionless scattering data for (1.1), and suchsolutions are explicitly expressed in terms of the two matrix triplets describingthe time-evolved bound-state data for (1.1).
In this section we introduce the Jost solutions and the scattering coefficientsfor each of the linear systems given in (1.1), (1.5), (1.6), and we present someof their relevant properties. For clarification, we use the superscript ( q, r ) todenote the quantities relevant to (1.1), use ( u, v ) for those relevant to (1.5), anduse ( p, s ) for those relevant to (1.6). When these three potential pairs decayrapidly in their respective equations as n → ±∞ , the corresponding coefficientmatrices all reduce to the same unperturbed coefficient matrix. In other words,each of (1.1), (1.5), (1.6) corresponds to the same unperturbed system˚Ψ n = z
00 1 z ˚Ψ n +1 , n ∈ Z , (cid:20) z − n (cid:21) and (cid:20) z n (cid:21) , i.e. we have˚Ψ n = a " z − n + b " z n , n ∈ Z , (2.1)with a and b being two complex-valued scalars that are independent of n butmay depend on z. There are four Jost solutions for each of (1.1), (1.5), (1.6), and they areobtained by assigning specific values to a and b as n → + ∞ or n → −∞ . Weuniquely define the four Jost solutions ψ n , φ n , ¯ ψ n , ¯ φ n to each of (1.1), (1.5),(1.6) so that they satisfy the respective asymptotics ψ n = " o (1) z n [1 + o (1)] , n → + ∞ , (2.2) φ n = " z − n [1 + o (1)] o (1) , n → −∞ , (2.3)¯ ψ n = " z − n [1 + o (1)] o (1) , n → + ∞ , (2.4)¯ φ n = " o (1) z n [1 + o (1)] , n → −∞ . (2.5)We remark that an overbar does not denote complex conjugation. We will usethe notation ψ ( q,r ) n , φ ( q,r ) n , ¯ ψ ( q,r ) n , ¯ φ ( q,r ) n to refer to the respective Jost solutionsfor (1.1); use ψ ( u,v ) n , φ ( u,v ) n , ¯ ψ ( u,v ) n , ¯ φ ( u,v ) n for the respective Jost solutions for(1.5); and use ψ ( p,s ) n , φ ( p,s ) n , ¯ ψ ( p,s ) n , ¯ φ ( p,s ) n for the respective Jost solutions for(1.6).The asymptotics of the Jost solutions complementary to (2.2)–(2.5) are usedto define the corresponding scattering coefficients compatible with (2.1). Wehave ψ n = LT l z − n [1 + o (1)]1 T l z n [1 + o (1)] , n → −∞ , (2.6) φ n = T r z − n [1 + o (1)] RT r z n [1 + o (1)] , n → + ∞ , (2.7)¯ ψ n = T l z − n [1 + o (1)]¯ L ¯ T l z n [1 + o (1)] , n → −∞ , (2.8)6 φ n = ¯ R ¯ T r z − n [1 + o (1)]1¯ T r z n [1 + o (1)] , n → + ∞ , (2.9)where T l and ¯ T l are the transmission coefficients from the left, T r and ¯ T r are thetransmission coefficients from the right, R and ¯ R are the reflection coefficientsfrom the right, and L and ¯ L are the reflection coefficients from the left. We willalso say left scattering coefficients instead of scattering coefficients from the left,and similarly we will use right scattering coefficients and scattering coefficientsfrom the right interchangeably.Note that we will use T ( q,r )r , T ( q,r )l , R ( q,r ) , L ( q,r ) , ¯ T ( q,r )r , ¯ T ( q,r )l , ¯ R ( q,r ) , ¯ L ( q,r ) to refer to the scattering coefficients for (1.1); use T ( u,v )r , T ( u,v )l , R ( u,v ) , L ( u,v ) , ¯ T ( u,v )r , ¯ T ( u,v )l , ¯ R ( u,v ) , ¯ L ( u,v ) for the scattering coefficients for (1.5); and use T ( p,s )r , T ( p,s )l , R ( p,s ) , L ( p,s ) , ¯ T ( p,s )r , ¯ T ( p,s )l , ¯ R ( p,s ) , ¯ L ( p,s ) for the scattering coeffi-cients for (1.6).Related to the linear system (1.5), let us introduce the quantities D ( u,v ) n and D ( u,v ) ∞ as D ( u,v ) n := n Y j = −∞ (1 − u j v j ) , D ( u,v ) ∞ := ∞ Y j = −∞ (1 − u j v j ) . (2.10)From the fact that u n and v n are rapidly decaying and that 1 − u n v n = 0 for n ∈ Z , it follows that D ( u,v ) n and D ( u,v ) ∞ are each well defined and nonzero.Similarly, related to the linear system (1.6), we let D ( p,s ) n := n Y j = −∞ (1 − p j s j ) , D ( p,s ) ∞ := ∞ Y j = −∞ (1 − p j s j ) . (2.11)From the fact that p n and s n are decaying rapidly and that 1 − p n s n = 0 for n ∈ Z , we see that D ( p,s ) n and D ( p,s ) ∞ are each well defined and nonzero.In the next theorem we list some relevant analyticity properties of the Jostsolutions to (1.5). Theorem 2.1.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and − u n v n = 0 for n ∈ Z . Then, the corresponding Jostsolutions to (1.5) satisfy following: (a) For each n ∈ Z the quantities z − n ψ ( u,v ) n , z n φ ( u,v ) n , z n ¯ ψ ( u,v ) n , z − n ¯ φ ( u,v ) n are even in z in their respective domains. (b) The quantity z − n ψ ( u,v ) n is analytic in | z | < and continuous in | z | ≤ . (c) The quantity z n φ ( u,v ) n is analytic in | z | < and continuous in | z | ≤ . (d) The quantity z n ¯ ψ ( u,v ) n is analytic in | z | > and continuous in | z | ≥ . The quantity z − n ¯ φ ( u,v ) n is analytic in | z | > and continuous in | z | ≥ . (f) The Jost solution ψ ( u,v ) n has the expansion ψ ( u,v ) n = ∞ X l = n K ( u,v ) nl z l , | z | ≤ , (2.12) with the double-indexed quantities K ( u,v ) nl for which we have K ( u,v ) nn = " , K ( u,v ) n ( n +2) = u n ∞ X k = n u k +1 v k , (2.13) and that K ( u,v ) nl = 0 when n + l is odd or l < n . (g) The Jost solution ¯ ψ ( u,v ) n has the expansion ¯ ψ ( u,v ) n = ∞ X l = n ¯ K ( u,v ) nl z l , | z | ≥ , (2.14) with the double-indexed quantities ¯ K ( u,v ) nl for which we have ¯ K ( u,v ) nn = " , ¯ K ( u,v ) n ( n +2) = ∞ X k = n u k v k +1 v n , (2.15) and that ¯ K ( u,v ) nl = 0 when n + l is odd or l < n . (h) For the Jost solution φ ( u,v ) n we have the expansion z n φ ( u,v ) n = ∞ X l =0 P ( u,v ) nl z l , | z | ≤ , (2.16) with the double-indexed quantities P ( u,v ) nl for which we have P ( u,v ) n = 1 D ( u,v ) n − " − v n − , (2.17) P ( u,v ) n = 1 D ( u,v ) n − n − X k = −∞ u k +1 v k − v n − − v n − n − X k = −∞ u k +1 v k , with D ( u,v ) n − being the quantity defined in (2.10) and that P ( u,v ) nl = 0 when l is odd or l < . For the Jost solution ¯ φ ( u,v ) n we have the expansion z − n ¯ φ ( u,v ) n = ∞ X l =0 ¯ P ( u,v ) nl z l , | z | ≥ , with the double-indexed quantities ¯ P ( u,v ) nl for which we have ¯ P ( u,v ) n = 1 D ( u,v ) n − " − u n − , ¯ P ( u,v ) n = 1 D ( u,v ) n − − u n − − u n − n − X k = −∞ u k v k +1 n − X k = −∞ u k v k +1 , and that ¯ P ( u,v ) nl = 0 when l is odd or l < . (j) The scattering coefficients for (1.5) are even in z in their respective do-mains. The domain for the reflection coefficients is the unit circle T andthe domains for the transmission coefficients consist of the union of T andtheir regions of extensions. (k) The quantities /T ( u,v )l and /T ( u,v )r have analytic extensions in z from z ∈ T to | z | < and those extensions are continuous for | z | ≤ . Similarly,the quantities / ¯ T ( u,v )l and / ¯ T ( u,v )r have extensions from z ∈ T so thatthey are analytic in | z | > and continuous in | z | ≥ . Proof.
We can write (1.5) for ψ ( u,v ) n in the equivalent form z − n ψ ( u,v ) n = " z z u n v n z − n − ψ ( u,v ) n +1 , n ∈ Z . (2.18)From (2.2) and the iteration of (2.18) in n, it follows that z − n ψ ( u,v ) n is aneven function of z. By proceeding in a similar manner for the remaining Jostsolutions, we complete the proof of (a). The expansion of z − n ψ ( u,v ) n obtained in(a) contains only nonnegative integer powers of z and is uniformly convergentin z for | z | ≤ , from which we conclude (b) and (f). The proofs for (c), (d), (e),(g), (h), (i) are obtained in a similar manner. Using (a)–(e) in (2.6)–(2.9) weestablish (j). Finally, using (b) and the second component of the column-vectorasymptotic in (2.6), we establish (k) for 1 /T ( u,v )l . The remaining proofs for (k)are obtained in a similar manner.We remark that the results in Theorem 2.1 holds also for (1.6). For theconvenience of citing those results, we present the following corollary.9 orollary 2.2.
Assume that the potentials p n and s n appearing in (1.6) arerapidly decaying and − p n s n = 0 for n ∈ Z . Then, the corresponding Jostsolutions to (1.6) satisfy all the properties stated in Theorem 2.1. In particularwe have the following: (a) The Jost solution ψ ( p,s ) n has the expansion ψ ( p,s ) n = ∞ X l = n K ( p,s ) nl z l , | z | ≤ , (2.19) with the double-indexed quantities K ( p,s ) nl for which we have K ( p,s ) nn = " , K ( p,s ) n ( n +2) = p n ∞ X k = n p k +1 s k , (2.20) and that K ( p,s ) nl = 0 when n + l is odd or l < n . (b) The Jost solution ¯ ψ ( p,s ) n has the expansion ¯ ψ ( p,s ) n = ∞ X l = n ¯ K ( p,s ) nl z l , | z | ≥ , (2.21) with the double-indexed quantities ¯ K ( p,s ) nl for which we have ¯ K ( p,s ) nn = " , ¯ K ( p,s ) n ( n +2) = ∞ X k = n p k s k +1 s n , (2.22) and that ¯ K ( p,s ) nl = 0 when n + l is odd or l < n . In the next theorem we summarize the relevant properties of the scatteringcoefficients for (1.5).
Theorem 2.3.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and that − u n v n = 0 for n ∈ Z . The corresponding scatteringcoefficients in their respective domains satisfy T ( u,v )r = D ( u,v ) ∞ T ( u,v )l , ¯ T ( u,v )r = D ( u,v ) ∞ ¯ T ( u,v )l , (2.23) T ( u,v )r ¯ T ( u,v )r = D ( u,v ) ∞ (cid:2) − R ( u,v ) ¯ R ( u,v ) (cid:3) , (2.24) T ( u,v )l ¯ T ( u,v )l = D ( u,v ) ∞ (cid:2) − L ( u,v ) ¯ L ( u,v ) (cid:3) , (2.25) L ( u,v ) T ( u,v )l = − D ( u,v ) ∞ ¯ R ( u,v ) ¯ T ( u,v )r , (2.26)10 L ( u,v ) ¯ T ( u,v )l = − D ( u,v ) ∞ R ( u,v ) T ( u,v )r , (2.27) where D ( u,v ) ∞ is the quantity defined in (2.10) .Proof. From (1.5) we get the matrix equations h φ ( u,v ) n ψ ( u,v ) n i = z z u n z v n z h φ ( u,v ) n +1 ψ ( u,v ) n +1 i , n ∈ Z , (2.28) h ¯ ψ ( u,v ) n ¯ φ ( u,v ) n i = z z u n z v n z h ¯ ψ ( u,v ) n +1 ¯ φ ( u,v ) n +1 i , n ∈ Z . (2.29)Using iteration on the determinants of both sides of (2.28) and (2.29), respec-tively, for any pair of integers n and m with m > n we getdet h φ ( u,v ) n ψ ( u,v ) n i = (1 − u n v n ) · · · (1 − u m v m ) det h φ ( u,v ) m +1 ψ ( u,v ) m +1 i , (2.30)det h ¯ ψ ( u,v ) n ¯ φ ( u,v ) n i = (1 − u n v n ) · · · (1 − u m v m ) det h ¯ ψ ( u,v ) m +1 ¯ φ ( u,v ) m +1 i . (2.31)Letting n → −∞ and m → + ∞ in (2.30), with the help of (2.2), (2.3), (2.6),(2.7), and (2.10) we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( u,v ) T ( u,v )l T ( u,v )l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = D ( u,v ) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ( u,v )r R ( u,v ) T ( u,v )r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.32)Similarly, letting n → −∞ and m → + ∞ in (2.31) and using (2.4), (2.5), (2.8),(2.9), and (2.10) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ( u,v )l L ( u,v ) ¯ T ( u,v )l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = D ( u,v ) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ( u,v ) ¯ T ( u,v )r T ( u,v )r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.33)From (2.32) and (2.33) we get (2.23). On the other hand, with the help ofTheorem 2.1 we conclude that any two of the four Jost solutions ψ ( u,v ) n , φ ( u,v ) n , ¯ ψ ( u,v ) n , ¯ φ ( u,v ) n form a linearly independent set of solutions to (1.5) when z is onthe unit circle T . We can express φ ( u,v ) n and ¯ φ ( u,v ) n as linear combinations of ψ ( u,v ) n and ¯ ψ ( u,v ) n in a matrix form as φ ( u,v ) n ¯ φ ( u,v ) n = T ( u,v )r R ( u,v ) T ( u,v )r ¯ R ( u,v ) ¯ T ( u,v )r T ( u,v )r ¯ ψ ( u,v ) n ψ ( u,v ) n , z ∈ T , (2.34)11here the entries in the coefficient matrix are obtained with the help of (2.2),(2.4), (2.7), (2.9) for the Jost solutions to (1.5). In a similar way, with the helpof (2.3), (2.5), (2.6), (2.8) for the Jost solutions to (1.5) we get ¯ ψ ( u,v ) n ψ ( u,v ) n = T ( u,v )l ¯ L ( u,v ) ¯ T ( u,v )l L ( u,v ) T ( u,v )l T ( u,v )l φ ( u,v ) n ¯ φ ( u,v ) n , z ∈ T . (2.35)For the compatibility of (2.34) and (2.35) we must have T ( u,v )r R ( u,v ) T ( u,v )r ¯ R ( u,v ) ¯ T ( u,v )r T ( u,v )r T ( u,v )l ¯ L ( u,v ) ¯ T ( u,v )l L ( u,v ) T ( u,v )l T ( u,v )l = " , z ∈ T . (2.36)Then, using (2.23) and (2.36) we obtain (2.24)–(2.27).The above theorem indicates that the set of left scattering coefficients T ( u,v )l , ¯ T ( u,v )l , L ( u,v ) , ¯ L ( u,v ) can be expressed in terms of the set of right scatteringcoefficients T ( u,v )r , ¯ T ( u,v )r , R ( u,v ) , ¯ R ( u,v ) , and vice versa.In the next proposition we provide the asymptotics of the transmission co-efficients for (1.5). Proposition 2.4.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and that − u n v n = 0 for n ∈ Z . Then, the transmissioncoefficients for (1.5) have their asymptotics given by T ( u,v )l = 1 − z ∞ X k = −∞ u k +1 v k + O (cid:0) z (cid:1) , z → , (2.37) T ( u,v )r = D ( u,v ) ∞ (cid:20) − z ∞ X k = −∞ u k +1 v k + O (cid:0) z (cid:1) (cid:21) , z → , (2.38)¯ T ( u,v )l = 1 − z ∞ X k = −∞ u k v k +1 + O (cid:18) z (cid:19) , z → ∞ , (2.39)¯ T ( u,v )r = D ( u,v ) ∞ (cid:20) − z ∞ X k = −∞ u k v k +1 + O (cid:18) z (cid:19) (cid:21) , z → ∞ , (2.40) where D ( u,v ) ∞ is the quantity defined in (2.10) . roof. From Theorem 2.1(k) we know that 1 /T ( u,v )l and 1 /T ( u,v )r are analyticin | z | < / ¯ T ( u,v )l and 1 / ¯ T ( u,v )r are analytic in | z | > . Premultiply-ing both sides of (2.12) by z − n [0 1] , then letting n → −∞ in the resultingequation, and using (2.6) with ψ ( u,v ) n and (2.13) we obtain (2.37). Similarly,premultiplying both sides of (2.14) by z n [1 0] , then letting n → −∞ in theresulting equation, and using (2.8) with ¯ ψ ( u,v ) n and (2.15) we obtain (2.39).Finally, with the help of (2.23), (2.37), (2.39) we get (2.38) and (2.40).In the next theorem we provide various other relevant properties of thetransmission coefficients for (1.5). Theorem 2.5.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and that − u n v n = 0 for n ∈ Z . Then, for the transmissioncoefficients of (1.5) we have the following: (a) None of T ( u,v )l , T ( u,v )r , ¯ T ( u,v )l , ¯ T ( u,v )r can vanish when z ∈ T . (b) We have T ( u,v )l (0) = 1 , T ( u,v )r (0) = 1 D ( u,v ) ∞ = 0 , (2.41)1¯ T ( u,v )l ( ∞ ) = 1 , T ( u,v )r ( ∞ ) = 1 D ( u,v ) ∞ = 0 . (c) The quantity /T ( u,v )l has at most a finite number of zeros in < | z | < and the multiplicity of each such zero is finite. The zeros of /T ( u,v )r andthe multiplicities of those zeros are the same as those for /T ( u,v )l . (d) The quantity / ¯ T ( u,v )l has at most a finite number of zeros in | z | > andthe multiplicity of each such zero is finite. The zeros of / ¯ T ( u,v )r and themultiplicities of those zeros are the same as those for / ¯ T ( u,v )l . (e) The quantities T ( u,v )l and T ( u,v )r are meromorphic in | z | < . The numberof their poles and the multiplicities of those poles are both finite. Simi-larly, the quantities ¯ T ( u,v )l and ¯ T ( u,v )r are meromorphic in | z | > , and thenumber of their poles and the multiplicities of those poles are both finite. (f) If z j is a pole of T ( u,v )l and T ( u,v )r in < | z | < , then − z j is also a poleof those two transmission coefficients. Similarly, if ¯ z j is a pole of ¯ T ( u,v )l and ¯ T ( u,v )r in | z | > then − ¯ z j is also a pole of ¯ T ( u,v )l and ¯ T ( u,v )r . Proof.
We can write (2.25) as1 D ( u,v ) ∞ = 1 T ( u,v )l ¯ T ( u,v )l − L ( u,v ) ¯ L ( u,v ) T ( u,v )l ¯ T ( u,v )l , z ∈ T . L ( u,v ) /T ( u,v )l , we would conclude that if T ( u,v )l vanished at some point on T then L ( u,v ) would have to vanish at that samepoint on T . However, this cannot happen because it would contradict (2.25) aswe have D ( u,v ) ∞ = 0 . The remaining proofs in (a) are established in a similarmanner. Note that (b) is obtained directly from (2.37)–(2.40). The proof of (c)for T ( u,v )l can be given as follows. From Theorem 2.1(k) we know that 1 /T ( u,v )l is analytic in | z | < | z | ≤
1, and from (a) we know that1 /T ( u,v )l cannot vanish on T . Hence, any zeros of 1 /T ( u,v )l must occur in thebounded region | z | < . Thus, the zeros of 1 /T ( u,v )l in | z | < u, v ) in Proposition 2.4 and Theorems 2.1, 2.3, and 2.5 are also valid for thepotential pair ( p, s ).Next, let us consider the properties of the scattering coefficients for (1.1).Because the coefficient matrix in (1.1) has determinant equal to 1 , in this casewe can express the scattering coefficients for (1.1) in terms of the Wronskianseach defined as a determinant of a 2 × (cid:20) α n β n (cid:21) and (cid:20) ˆ α n ˆ β n (cid:21) to (1.1) as "" α n β n ; " ˆ α n ˆ β n := det " α n ˆ α n β n ˆ β n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α n ˆ α n β n ˆ β n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.42)We recall that the scattering coefficients for (1.5) cannot be obtained fromthe Wronskians of any two solutions to (1.5) because the coefficient matrix in(1.5) does not have the determinant equal to 1 . In that case, in order to obtainthe scattering coefficients one can use the Wronskians of a solution to (1.5) anda solution to the adjoint equation corresponding to (1.5). However, we preferto express the scattering coefficients via the asymptotics of the Jost solutionsas in (2.6)–(2.9) and this allows us to investigate the scattering coefficients in aunified way for any of the three systems (1.1), (1.5), (1.6).With the help of (1.1) and (2.42) one can directly verify that the determinantused in (2.42) is independent of n . In terms of the Jost solutions ψ ( q,r ) n , φ ( q,r ) n , ¯ ψ ( q,r ) n , ¯ φ ( q,r ) n satisfying (1.1) and the respective asymptotics given in (2.2)–(2.5),with the help of (2.6)–(2.9) we express the scattering coefficients T ( q,r )l , ¯ T ( q,r )l ,T ( q,r )r , ¯ T ( q,r )r , R ( q,r ) , ¯ R ( q,r ) , L ( q,r ) , ¯ L ( q,r ) as1 T ( q,r )l = (cid:12)(cid:12)(cid:12) φ ( q,r ) n ψ ( q,r ) n (cid:12)(cid:12)(cid:12) , T ( q,r )l = (cid:12)(cid:12)(cid:12) ¯ ψ ( q,r ) n ¯ φ ( q,r ) n (cid:12)(cid:12)(cid:12) , (2.43)14 T ( q,r )r = (cid:12)(cid:12)(cid:12) φ ( q,r ) n ψ ( q,r ) n (cid:12)(cid:12)(cid:12) , T ( q,r )r = (cid:12)(cid:12)(cid:12) ¯ ψ ( q,r ) n ¯ φ ( q,r ) n (cid:12)(cid:12)(cid:12) , (2.44) L ( q,r ) T ( q,r )l = (cid:12)(cid:12)(cid:12) ψ ( q,r ) n ¯ φ ( q,r ) n (cid:12)(cid:12)(cid:12) , ¯ L ( q,r ) ¯ T ( q,r )l = (cid:12)(cid:12)(cid:12) φ ( q,r ) n ¯ ψ ( q,r ) n (cid:12)(cid:12)(cid:12) , (2.45) R ( q,r ) T ( q,r )r = (cid:12)(cid:12)(cid:12) ¯ ψ ( q,r ) n φ ( q,r ) n (cid:12)(cid:12)(cid:12) , ¯ R ( q,r ) ¯ T ( q,r )r = (cid:12)(cid:12)(cid:12) ¯ φ ( q,r ) n ψ ( q,r ) n (cid:12)(cid:12)(cid:12) . (2.46)In the next theorem we list some relevant properties of the scattering coef-ficients for (1.1). Theorem 2.6.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, we have the following: (a) The left and right transmission coefficients for (1.1) are equal to eachother, i.e. we have T ( q,r )l = T ( q,r )r , ¯ T ( q,r )l = ¯ T ( q,r )r . (2.47) We will use T ( q,r ) to denote the common value of T ( q,r )l and T ( q,r )r , andwe will use ¯ T ( q,r ) to denote the common value of ¯ T ( q,r )l and ¯ T ( q,r )r . Thetransmission coefficient T ( q,r ) has a meromorphic extension from z ∈ T to | z | < and the transmission coefficient ¯ T ( q,r ) has a meromorphic ex-tension from z ∈ T to | z | > . (b) For z ∈ T , the left and right reflection coefficients for (1.1) satisfy L ( q,r ) T ( q,r ) = − ¯ R ( q,r ) ¯ T ( q,r ) , ¯ L ( q,r ) ¯ T ( q,r ) = − R ( q,r ) T ( q,r ) , (2.48) T ( q,r ) ¯ T ( q,r ) = 1 − L ( q,r ) ¯ L ( q,r ) = 1 − R ( q,r ) ¯ R ( q,r ) . (2.49) Proof.
The proof can be obtained as in the proof of Theorem 2.3. As an alternateproof, we see that (2.47) follows from (2.43) and (2.44); (2.48) follows from(2.45) and (2.46); and that (2.49) is established by using the fact that theWronskian of ¯ ψ ( q,r ) n and ψ ( q,r ) n as in (2.42) yields the same value as n → −∞ and n → + ∞ . Finally, the aforementioned meromorphic extensions for thetransmission coefficients follow from the fact that the Jost solutions ψ ( q,r ) n and φ ( q,r ) n have analytic extensions in z to | z | < ψ ( q,r ) n and ¯ φ ( q,r ) n have analytic extensions in z to | z | > , where the analytic extensionscan be established by iterating (1.1) and by using (2.2)–(2.5).We see from (2.47) that the left and right transmission coefficients for (1.1)are equal whereas the same does not hold for (1.5). Similar to (2.10) we definethe related quantities D ( q,r ) n and D ( q,r ) ∞ for (1.1) as D ( q,r ) n := n Y j = −∞ (1 − q j r j ) , D ( q,r ) ∞ := ∞ Y j = −∞ (1 − q j r j ) . (2.50)15or (1.1) we also define the quantities E ( q,r ) n and E ( q,r ) ∞ as E ( q,r ) n := n Y j = −∞ (1 + q j r j +1 ) , E ( q,r ) ∞ := ∞ Y j = −∞ (1 + q j r j +1 ) . (2.51)From (1.2) and the fact that q n and r n decay rapidly, it follows that the quan-tities D ( q,r ) n , D ( q,r ) ∞ , E ( q,r ) n , D ( q,r ) ∞ are each well defined and nonzero.Let us introduce the scalar quantities S ( q,r ) n and Q ( q,r ) n as S ( q,r ) n := n X k = −∞ r k ( q k − q k +1 − q k q k +1 r k +1 )(1 − q k r k )(1 − q k +1 r k +1 ) , (2.52) Q ( q,r ) n := n X k = −∞ r k +2 ( q k − q k +1 − q k q k +1 r k +1 )(1 + q k r k +1 )(1 + q k +1 r k +2 ) . (2.53)Letting n → + ∞ in (2.52) and (2.53) we get S ( q,r ) ∞ := ∞ X k = −∞ r k ( q k − q k +1 − q k q k +1 r k +1 )(1 − q k r k )(1 − q k +1 r k +1 ) , (2.54) Q ( q,r ) ∞ := ∞ X k = −∞ r k +2 ( q k − q k +1 − q k q k +1 r k +1 )(1 + q k r k +1 )(1 + q k +1 r k +2 ) . (2.55)As stated in Theorem 2.6(a), the transmission coefficient T ( q,r ) for (1.1) hasa meromorphic extension to | z | < T ( q,r ) for(1.1) has a meromorphic extension to | z | > . The asymptotics of those twotransmission coefficients are presented next.
Proposition 2.7.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, the small- z asymptotics of transmis-sion coefficient T ( q,r ) for (1.1) is given by T ( q,r ) = 1 D ( q,r ) ∞ h − z S ( q,r ) ∞ + O (cid:0) z (cid:1)i , z → , (2.56) where D ( q,r ) ∞ and S ( q,r ) ∞ are the scalar quantities defined in (2.50) and (2.54) ,respectively. Similarly, the large- z asymptotics of transmission coefficient ¯ T ( q,r ) for (1.1) is given by ¯ T ( q,r ) = 1 E ( q,r ) ∞ (cid:20) − z Q ( q,r ) ∞ + O (cid:18) z (cid:19)(cid:21) , z → ∞ , (2.57) where E ( q,r ) ∞ and Q ( q,r ) ∞ are the scalar quantities defined in (2.51) and (2.55) ,respectively. roof. The proof is lengthy but straightforward. To obtain (2.56) we use (1.1)with the Jost solution ψ ( q,r ) n , premultiply both sides of (1.1) with z − n [0 1] , iterate the resulting equation, and for n < m we get (cid:2) (cid:3) z − n ψ ( q,r ) n = (cid:2) (cid:3) Ξ n Ξ n +1 · · · Ξ m z − m − ψ ( q,r ) m +1 , (2.58)where we have definedΞ n := " − q n − q n r n + z " q n r n q n r n . We note that in the limit n → −∞ the left-hand side of (2.58) yields 1 /T ( q,r ) .Letting n → −∞ and m → + ∞ in (2.58), using (2.2) and (2.6) for ψ ( q,r ) n and also using D ( q,r ) ∞ defined in (2.50) and S ( q,r ) ∞ defined in (2.54), after somestraightforward algebra we get (2.56). The proof of (2.57) is similarly obtainedby using (1.1) with the Jost solution ¯ ψ ( q,r ) n , premultiplying both sides of (1.1)with z n [1 0] , iterating the resulting equation, and using (2.4) and (2.8) for¯ ψ ( q,r ) n and also using E ( q,r ) ∞ defined in (2.51) and Q ( q,r ) ∞ defined in (2.55).In the next theorem we provide some further relevant properties of the trans-mission coefficients for (1.1). Theorem 2.8.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, for the transmission coefficients T ( q,r ) and ¯ T ( q,r ) of (1.1) we have the following: (a) Neither T ( q,r ) nor ¯ T ( q,r ) can vanish when z ∈ T . (b) We have T ( q,r ) (0) = D ( q,r ) ∞ , T ( q,r ) ( ∞ ) = E ( q,r ) ∞ . (2.59)(c) The quantity /T ( q,r ) has at most a finite number of zeros in < | z | < and the multiplicity of each such zero is finite. (d) The quantity / ¯ T ( q,r ) has at most a finite number of zeros in | z | > andthe multiplicity of each such zero is finite. (e) The transmission coefficient T ( q,r ) is meromorphic in | z | < , and thenumber of its poles and the multiplicities of those poles are both finite.Similarly, ¯ T ( q,r ) is meromorphic in | z | > , and the number of its polesand the multiplicities of those poles are both finite. (f) If z j is a pole of T ( q,r ) in < | z | < , then − z j is also a pole of T ( q,r ) .Similarly, if ¯ z j is a pole of ¯ T ( q,r ) in | z | > , then − ¯ z j is also a pole of ¯ T ( q,r ) . Proof.
We note that (2.59) follows from (2.56) and (2.57). The rest of the proofcan be given in a way similar to the proof of Theorem 2.5.17inally, in this section we clarify the relationship between the poles of thetransmission coefficients and the linear dependence of the relevant Jost solutionsfor each of the linear systems (1.1), (1.5), and (1.6). This clarification has manyimportant consequences. It allows us to introduce the dependency constants atthe bound states. It also allows us to deal with bound states of any multiplicitiesin an elegant manner. The treatment given here for the linear systems (1.1),(1.5), and (1.6) can be readily generalized to other linear systems both in thediscrete and continuous cases.In terms of the Jost solutions ψ n , φ n , ¯ ψ n , ¯ φ n appearing in (2.2)–(2.5) foreach of the linear systems (1.1), (1.5), and (1.6), we define a ( q,r ) n := (cid:12)(cid:12)(cid:12) φ ( q,r ) n ψ ( q,r ) n (cid:12)(cid:12)(cid:12) , ¯ a ( q,r ) n := (cid:12)(cid:12)(cid:12) ¯ φ ( q,r ) n ¯ ψ ( q,r ) n (cid:12)(cid:12)(cid:12) , (2.60) a ( u,v ) n := (cid:12)(cid:12)(cid:12) φ ( u,v ) n ψ ( u,v ) n (cid:12)(cid:12)(cid:12) , ¯ a ( u,v ) n := (cid:12)(cid:12)(cid:12) ¯ φ ( u,v ) n ¯ ψ ( u,v ) n (cid:12)(cid:12)(cid:12) , (2.61) a ( p,s ) n := (cid:12)(cid:12)(cid:12) φ ( p,s ) n ψ ( p,s ) n (cid:12)(cid:12)(cid:12) , ¯ a ( p,s ) n := (cid:12)(cid:12)(cid:12) ¯ φ ( p,s ) n ¯ ψ ( p,s ) n (cid:12)(cid:12)(cid:12) , (2.62)where on the right-hand sides we have the Wronskian determinants.The relationships among a ( q,r ) n , ¯ a ( q,r ) n , and the transmission coefficients T ( q,r ) and ¯ T ( q,r ) for (1.1) are clarified in the following theorem. Theorem 2.9.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, we have the following: (a) The scalar quantities a ( q,r ) n and ¯ a ( q,r ) n defined in (2.60) are independent of n, depend only on z, and are related to the transmission coefficients T ( q,r ) and ¯ T ( q,r ) appearing in (2.43) , (2.44) , (2.47) as a ( q,r ) n = 1 T ( q,r ) , ¯ a ( q,r ) n = − T ( q,r ) . (2.63)(b) Consequently, the linear dependence of the Jost solutions φ ( q,r ) n and ψ ( q,r ) n occurs at the poles of T ( q,r ) in < | z | < . Similarly, the linear dependenceof the Jost solutions ¯ φ ( q,r ) n and ¯ ψ ( q,r ) n occurs at the poles of ¯ T ( q,r ) in | z | > . (c) In particular, if T ( q,r ) has a pole at z = ± z j with multiplicity m j , then wehave d k a ( q,r ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± z j = 0 , ≤ k ≤ m j − , n ∈ Z . (2.64) Similarly, if ¯ T ( q,r ) has a pole at z = ± ¯ z j with multiplicity ¯ m j , we thenhave d k ¯ a ( q,r ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± ¯ z j = 0 , ≤ k ≤ ¯ m j − , n ∈ Z . (2.65)18 roof. Note that (2.63) is obtained directly from (2.43), (2.44), (2.47), and(2.60). Since each of ψ ( q,r ) n , φ ( q,r ) n , ¯ ψ ( q,r ) n , ¯ φ ( q,r ) n satisfies the same linear system(1.1), the linear dependence and the vanishing of the Wronskian determinantare equivalent. We also note that (2.64) and (2.65) directly follow (2.63).In the next theorem we clarify the relationships among a ( u,v ) n , ¯ a ( u,v ) n , and thetransmission coefficients for (1.5). Theorem 2.10.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and that − u n v n = 0 for n ∈ Z . Then, we have the following: (a) The scalar quantities a ( u,v ) n and ¯ a ( u,v ) n defined in (2.61) depend both on n and z, and they are related to the transmission coefficients T ( u,v )r and ¯ T ( u,v )r appearing in (2.23) and (2.24) as a ( u,v ) n = D ( u,v ) ∞ D ( u,v ) n − T ( u,v )r , ¯ a ( u,v ) n = − D ( u,v ) ∞ D ( u,v ) n − T ( u,v )r , (2.66) where D ( u,v ) n and D ( u,v ) ∞ are the scalar quantities defined in (2.10) . (b) Since D ( u,v ) ∞ = 0 and D ( u,v ) n = 0 for n ∈ Z and these quantities do notcontain z, we conclude from (2.61) and (2.66) that the linear dependenceof the Jost solutions φ ( u,v ) n and ψ ( u,v ) n occurs at the poles of T ( u,v )r and thatthe linear dependence of the Jost solutions ¯ φ ( u,v ) n and ¯ ψ ( u,v ) n occurs at thepoles of ¯ T ( u,v )r . (c) In particular, if T ( u,v )r has a pole at z = ± z j of multiplicity m j , then wehave d k a ( u,v ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± z j = 0 , ≤ k ≤ m j − , n ∈ Z . (2.67) Similarly, if ¯ T ( u,v )r has a pole at z = ± ¯ z j of multiplicity ¯ m j , then we have d k ¯ a ( u,v ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± ¯ z j = 0 , ≤ k ≤ ¯ m j − , n ∈ Z . (2.68) Proof.
Let us use X n and |X n | to denote the coefficient matrix in (1.5) and itsdeterminant, respectively, i.e. X n := z z u n z v n z , |X n | := 1 − u n v n . (2.69)From (1.5) and (2.61) we get a ( u,v ) n = (cid:12)(cid:12)(cid:12) X n φ ( u,v ) n +1 X n ψ ( u,v ) n +1 (cid:12)(cid:12)(cid:12) ,
19r equivalently a ( u,v ) n = (cid:12)(cid:12) X n (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) φ ( u,v ) n +1 ψ ( u,v ) n +1 (cid:12)(cid:12)(cid:12) . Iterating in this manner, from (1.5) and (2.61), for m > n we obtain a ( u,v ) n = (cid:12)(cid:12) X n (cid:12)(cid:12) (cid:12)(cid:12) X n +1 (cid:12)(cid:12) · · · (cid:12)(cid:12) X m (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) φ ( u,v ) m +1 ψ ( u,v ) m +1 (cid:12)(cid:12)(cid:12) . (2.70)Letting m → + ∞ in (2.70) and using (2.2) and (2.7) for the potential pair ( u, v ) , with the help of (2.69) we get a ( u,v ) n = ∞ Y k = n (1 − u k v k ) ! T ( u,v )r , which is equivalent to a ( u,v ) n = ∞ Y k = −∞ (1 − u k v k ) n − Y k = −∞ (1 − u k v k ) 1 T ( u,v )r . (2.71)Using (2.10) in (2.71) we get the first equality in (2.66). The second equality in(2.66) is obtained by iterating (1.5) as (cid:12)(cid:12)(cid:12) ¯ φ ( u,v ) n ¯ ψ ( u,v ) n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) X n (cid:12)(cid:12) (cid:12)(cid:12) X n +1 (cid:12)(cid:12) · · · (cid:12)(cid:12) X m (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ¯ φ ( u,v ) m +1 ¯ ψ ( u,v ) m +1 (cid:12)(cid:12)(cid:12) , and letting m → + ∞ and using (2.4) and (2.9) for the potential pair ( u, v ). Wefinally remark that (b) and (c) are direct consequences of (a).The result of Theorem 2.10 is remarkable in the sense that, even though thescattering coefficients for (1.5) cannot be defined as some Wronskians of Jostsolutions to (1.5) as in (2.43)–(2.46), we have the relations given by1 T ( u,v )r = D ( u,v ) n − D ( u,v ) ∞ (cid:12)(cid:12)(cid:12) φ ( u,v ) n ψ ( u,v ) n (cid:12)(cid:12)(cid:12) , T ( u,v )r = D ( u,v ) n − D ( u,v ) ∞ (cid:12)(cid:12)(cid:12) ¯ ψ ( u,v ) n ¯ φ ( u,v ) n (cid:12)(cid:12)(cid:12) , (2.72)1 T ( u,v )l = 1 D ( u,v ) n − (cid:12)(cid:12)(cid:12) φ ( u,v ) n ψ ( u,v ) n (cid:12)(cid:12)(cid:12) , T ( u,v )l = 1 D ( u,v ) n − (cid:12)(cid:12)(cid:12) ¯ ψ ( u,v ) n ¯ φ ( u,v ) n (cid:12)(cid:12)(cid:12) , (2.73) R ( u,v ) T ( u,v )r = D ( u,v ) n − D ( u,v ) ∞ (cid:12)(cid:12)(cid:12) ¯ ψ ( u,v ) n φ ( u,v ) n (cid:12)(cid:12)(cid:12) , ¯ R ( u,v ) ¯ T ( u,v )r = D ( u,v ) n − D ( u,v ) ∞ (cid:12)(cid:12)(cid:12) ¯ φ ( u,v ) n ψ ( u,v ) n (cid:12)(cid:12)(cid:12) , (2.74) L ( u,v ) T ( u,v )l = 1 D ( u,v ) n − (cid:12)(cid:12)(cid:12) ψ ( u,v ) n ¯ φ ( u,v ) n (cid:12)(cid:12)(cid:12) , ¯ L ( u,v ) ¯ T ( u,v )l = 1 D ( u,v ) n − (cid:12)(cid:12)(cid:12) φ ( u,v ) n ¯ ψ ( u,v ) n (cid:12)(cid:12)(cid:12) . (2.75)Note that (2.72) and (2.74) can be obtained by using a forward iteration as in(2.70) and that (2.73) and (2.75) can be obtained via a backward iteration on201.5). For example, the first equality in (2.73) is obtained by letting m → −∞ in (cid:12)(cid:12)(cid:12) φ ( u,v ) n ψ ( u,v ) n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) X − n − (cid:12)(cid:12) (cid:12)(cid:12) X − n − (cid:12)(cid:12) · · · (cid:12)(cid:12) X − m (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) φ ( u,v ) m ψ ( u,v ) m (cid:12)(cid:12)(cid:12) . Note that the scalar quantity D ( u,v ) ∞ in (2.72) is independent of z and nonzero.Furthermore, each D ( u,v ) n for n ∈ Z is independent of z and nonzero. Thus,(2.72) allows us to directly relate the poles of T ( u,v )r to the zeros of the Wronskiandeterminant | φ ( u,v ) n ψ ( u,v ) n | , and similarly we can directly relate the poles of¯ T ( u,v )r to the zeros of the Wronskian determinant | ¯ φ ( u,v ) n ¯ ψ ( u,v ) n | . The results stated in Theorem 2.10 hold for (1.6) as well. In the followingcorollary we state those results without a proof since that proof is similar to theproof of Theorem 2.10.
Corollary 2.11.
Assume that the potentials p n and s n appearing in (1.6) arerapidly decaying and that − p n s n = 0 for n ∈ Z . Then, we have the following: (a) The scalar quantities a ( p,s ) n and ¯ a ( p,s ) n defined in (2.62) depend both on n and z, and they are related to the transmission coefficients T ( p,s )r and ¯ T ( p,s )r appearing in (2.7) and (2.9) for the potential pair ( p, s ) as a ( p,s ) n = D ( p,s ) ∞ D ( p,s ) n − T ( p,s )r , ¯ a ( p,s ) n = − D ( p,s ) ∞ D ( p,s ) n − T ( p,s )r , (2.76) where D ( p,s ) n and D ( p,s ) ∞ are the scalar quantities defined in (2.11) . (b) Since D ( p,s ) ∞ = 0 and D ( p,s ) n = 0 for n ∈ Z and these quantities do notcontain z, we conclude from (2.62) and (2.76) that the linear dependenceof the Jost solutions φ ( p,s ) n and ψ ( p,s ) n occurs at the poles of T ( p,s )r and thatthe linear dependence of the Jost solutions ¯ φ ( p,s ) n and ¯ ψ ( p,s ) n occurs at thepoles of ¯ T ( p,s )r . (c) In particular, if T ( p,s )r has a pole at z = ± z j of multiplicity m j , then wehave d k a ( p,s ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± z j = 0 , ≤ k ≤ m j − , n ∈ Z . (2.77) Similarly, if ¯ T ( p,s )r has a pole at z = ± ¯ z j of multiplicity ¯ m j , then we have d k ¯ a ( p,s ) n dz k (cid:12)(cid:12)(cid:12)(cid:12) z = ± ¯ z j = 0 , ≤ k ≤ ¯ m j − , n ∈ Z . (2.78) In this section we relate the linear systems (1.1), (1.5), (1.6) to each other bychoosing the potential pairs ( u, v ) and ( p, s ) in terms of the potential pair ( q, r )21n a particular way, namely as u n = q n E ( q,r ) n − D ( q,r ) n , (3.1) v n = ( − r n + r n +1 − q n r n r n +1 ) D ( q,r ) n − E ( q,r ) n , (3.2) p n = ( q n − q n +1 − q n q n +1 r n +1 ) E ( q,r ) n − D ( q,r ) n +1 , (3.3) s n = r n +1 D ( q,r ) n E ( q,r ) n , (3.4)where D ( q,r ) n and E ( q,r ) n are the quantities defined in (2.50) and (2.51), respec-tively. This helps us to express the Jost solutions and the scattering coefficientsfor (1.5) and (1.6) in terms of the corresponding quantities for (1.1). In thissection we also present certain relevant properties of the Jost solutions to (1.1),and we express q n and r n in terms of the values at z = 1 of the Jost solutionsto (1.5) and (1.6). The results presented in this section play a crucial role insolving the direct and inverse scattering problems for (1.1) by exploiting thetechniques for the corresponding problems for (1.5) and (1.6).In the next proposition, when (3.1)–(3.4) are valid we present some relation-ships among the quantities for the potential pairs ( q, r ) , ( u, v ) , and ( p, s ) . Proposition 3.1.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Let the potential pairs ( u, v ) and ( p, s ) arerelated to ( q, r ) as in (3.1) – (3.4) . We then have − u n v n = 1(1 − q n r n )(1 + q n r n +1 ) , (3.5)1 − p n s n = 1(1 − q n +1 r n +1 )(1 + q n r n +1 ) , (3.6) D ( u,v ) n = 1 D ( q,r ) n E ( q,r ) n , D ( u,v ) ∞ = 1 D ( q,r ) ∞ E ( q,r ) ∞ , (3.7) D ( p,s ) n = 1 D ( q,r ) n +1 E ( q,r ) n , D ( p,s ) ∞ = 1 D ( q,r ) ∞ E ( q,r ) ∞ , (3.8) where we recall that D ( u,v ) n and D ( u,v ) ∞ are as in (2.10) , D ( p,s ) n and D ( p,s ) ∞ areas in (2.11) , D ( q,r ) n and D ( q,r ) ∞ are as in (2.50) , and E ( q,r ) n and E ( q,r ) ∞ are as in (2.51) .Proof. We evaluate the left-hand side of (3.5) with the help of (2.50), (2.51),(3.1), (3.2), and after a brief simplification we establish (3.5). Similarly, weobtain (3.6) with the help of (2.50), (2.51), (3.3), and (3.4). Then, we establish(3.7) by using (3.5) in (2.10), (2.50), and (2.51). Similarly, we get (3.8) by using(3.6) in (2.11), (2.50), and (2.51). 22he following proposition will be useful in solving the inverse problem for(1.1) by using the method introduced in (e) of Section 8.
Proposition 3.2.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Let the potential pairs ( u, v ) and ( p, s ) berelated to ( q, r ) as in (3.1) – (3.4) . Then, we have D ( q,r ) n = 1 n − Y k = −∞ (1 + u k +1 s k ) , (3.9) E ( q,r ) n = 1 n Y k = −∞ (1 − u k s k ) , (3.10) q n = u n n − Y k = −∞ − u k s k u k +1 s k , (3.11) r n = s n − n − Y k = −∞ u k s k − − u k s k , (3.12) where D ( q,r ) n and E ( q,r ) n are the scalar quantities defined in (2.50) and (2.51) ,respectively.Proof. From (3.1) and (3.4), we obtain1 − u n s n = 1 − q n E ( q,r ) n − D ( q,r ) n r n +1 D ( q,r ) n E ( q,r ) n , which, with the help of (2.51), simplifies to1 − u n s n = 11 + q n r n +1 . (3.13)Similarly, from (3.1) and (3.4) we get1 + u n s n − = 1 + q n E ( q,r ) n − D ( q,r ) n r n D ( q,r ) n − E ( q,r ) n − , which, with the help of (2.50), simplifies to1 + u n s n − = 11 − q n r n . (3.14)From (3.13) and (3.14), we respectively get n Y k = −∞ (1 − u k s k ) = 1 n Y k = −∞ (1 + q k r k +1 ) , Y k = −∞ (1 + u k s k − ) = 1 n Y k = −∞ (1 − q k r k ) , which yield (3.10) and (3.9), respectively. Finally, by using (3.9) and (3.10) in(3.1) and (3.4), we obtain (3.11) and (3.12), respectively.In the next theorem, when the potential pairs are related to each other as in(3.1)–(3.4), the Jost solutions to (1.1) are related to the Jost solutions to (1.5)and also to the Jost solutions to (1.6). Theorem 3.3.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) , and let the potential pairs ( u, v ) and ( p, s ) berelated to the potential pair ( q, r ) as in (3.1) – (3.4) . Then, the four Jost solutions ψ ( q,r ) n , φ ( q,r ) n , ¯ ψ ( q,r ) n , ¯ φ ( q,r ) n to (1.1) are related to the Jost solutions ψ ( u,v ) n , φ ( u,v ) n , ¯ ψ ( u,v ) n , ¯ φ ( u,v ) n to (1.5) and the Jost solutions ψ ( p,s ) n , φ ( p,s ) n , ¯ ψ ( p,s ) n , ¯ φ ( p,s ) n to (1.6) as ψ ( q,r ) n = D ( q,r ) ∞ (cid:18) − z (cid:19) E ( q,r ) n − r n E ( q,r ) n − D ( q,r ) n − ψ ( u,v ) n , (3.15) ψ ( q,r ) n = D ( q,r ) ∞ E ( q,r ) n − − q n D ( q,r ) n r n E ( q,r ) n − D ( q,r ) n − ψ ( p,s ) n , (3.16) φ ( q,r ) n = 11 − z (cid:18) − z (cid:19) E ( q,r ) n − r n E ( q,r ) n − D ( q,r ) n − φ ( u,v ) n , (3.17) φ ( q,r ) n = E ( q,r ) n − − q n D ( q,r ) n r n E ( q,r ) n − D ( q,r ) n − φ ( p,s ) n , (3.18)¯ ψ ( q,r ) n = E ( q,r ) ∞ − z (cid:18) − z (cid:19) E ( q,r ) n − r n E ( q,r ) n − D ( q,r ) n − ¯ ψ ( u,v ) n , (3.19)24 ψ ( q,r ) n = E ( q,r ) ∞ E ( q,r ) n − − q n D ( q,r ) n r n E ( q,r ) n − D ( q,r ) n − ¯ ψ ( p,s ) n , (3.20)¯ φ ( q,r ) n = (cid:18) − z (cid:19) E ( q,r ) n − r n E ( q,r ) n − D ( q,r ) n − ¯ φ ( u,v ) n , (3.21)¯ φ ( q,r ) n = E ( q,r ) n − − q n D ( q,r ) n r n E ( q,r ) n − D ( q,r ) n − ¯ φ ( p,s ) n , (3.22) where we recall that D ( q,r ) ∞ and E ( q,r ) ∞ are the constants defined in (2.50) and (2.51) , respectively.Proof. We only present the proof for (3.15) because the proofs for (3.16)–(3.22)can be obtained in a similar manner. To establish (3.15) we let ψ ( q,r ) n = Γ ( q,r ) n ψ ( u,v ) n , (3.23)where Γ ( q,r ) n is a 2 × ψ ( q,r ) n satisfies (1.1) and ψ ( u,v ) n satisfies (1.5), from (1.1), (1.5), (3.23) we obtain Γ ( q,r ) n as listed in (3.15).As an alternate proof we remark that the reader can directly verify that eachof (3.15)–(3.22) is compatible with (1.1), (1.5), (2.2)–(2.5), and (3.1)–(3.4).In the next theorem we relate the scattering coefficients for (1.1), (1.5), (1.6)to each other. Theorem 3.4.
Assume that the potential pair ( q, r ) is rapidly decaying andsatisfy (1.2) . Assume also that the potential pairs ( u, v ) and ( p, s ) are relatedto ( q, r ) as in (3.1) – (3.4) . Then, the scattering coefficients T ( q,r ) , ¯ T ( q,r ) , R ( q,r ) , ¯ R ( q,r ) , L ( q,r ) , ¯ L ( q,r ) for (1.1) are related to the scattering coefficients T ( u,v )l ,T ( u,v )r , ¯ T ( u,v )l , ¯ T ( u,v )r , R ( u,v ) , ¯ R ( u,v ) , L ( u,v ) , ¯ L ( u,v ) for (1.5) and T ( p,s )l , T ( p,s )r , ¯ T ( p,s )l , ¯ T ( p,s )r , R ( p,s ) , ¯ R ( p,s ) , L ( p,s ) , ¯ L ( p,s ) for (1.6) as T ( u,v )l = D ( q,r ) ∞ T ( q,r ) , T ( p,s )l = D ( q,r ) ∞ T ( q,r ) , (3.24) T ( u,v )r = 1 E ( q,r ) ∞ T ( q,r ) , T ( p,s )r = 1 E ( q,r ) ∞ T ( q,r ) , (3.25)¯ T ( u,v )l = E ( q,r ) ∞ ¯ T ( q,r ) , ¯ T ( p,s )l = E ( q,r ) ∞ ¯ T ( q,r ) , (3.26)¯ T ( u,v )r = 1 D ( q,r ) ∞ ¯ T ( q,r ) , ¯ T ( p,s )r = 1 D ( q,r ) ∞ ¯ T ( q,r ) , (3.27)25 ( u,v ) = (cid:18) − z (cid:19) D ( q,r ) ∞ E ( q,r ) ∞ R ( q,r ) , R ( p,s ) = D ( q,r ) ∞ E ( q,r ) ∞ R ( q,r ) , (3.28)¯ R ( u,v ) = 11 − z E ( q,r ) ∞ D ( q,r ) ∞ ¯ R ( q,r ) , ¯ R ( p,s ) = E ( q,r ) ∞ D ( q,r ) ∞ ¯ R ( q,r ) , (3.29) L ( u,v ) = 11 − z L ( q,r ) , L ( p,s ) = L ( q,r ) , (3.30)¯ L ( u,v ) = (cid:18) − z (cid:19) ¯ L ( q,r ) , ¯ L ( p,s ) = ¯ L ( q,r ) , (3.31) where we recall that D ( q,r ) ∞ and E ( q,r ) ∞ are the constants defined in (2.50) and (2.51) , respectively.Proof. We use the asymptotics of ψ ( q,r ) n , ψ ( u,v ) n , ψ ( p,s ) n given in (2.6) and we let n → −∞ in (3.15), which helps us to establish (3.24) and (3.25), respectively.We establish (3.26)–(3.31) in a similar manner.When (3.1)–(3.4) hold, from (2.41) and (3.24)–(3.27) we obtain the resultstated in the following corollary. Corollary 3.5.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Then, the transmission coeffi-cients T ( q,r ) , T ( u,v )l , T ( u,v )r , T ( p,s )l , T ( p,s )r have coinciding poles in < | z | < and the coinciding multiplicity for each of those poles. Similarly, ¯ T ( q,r ) , ¯ T ( u,v )l , ¯ T ( u,v )r , ¯ T ( p,s )l , ¯ T ( p,s )r have their coinciding poles in | z | > with the coincidingmultiplicity for each of those poles. When (3.1)–(3.4) hold, based on Corollary 3.5 we will use {± z j , m j } Nj =1 todenote the common set of poles in 0 < | z | < T ( q,r ) ,T ( u,v )l , T ( u,v )r , T ( p,s )l , T ( p,s )r , and similarly we will use {± ¯ z j , ¯ m j } ¯ Nj =1 to denotethe common set of poles in | z | > T ( q,r ) , ¯ T ( u,v )l , ¯ T ( u,v )r , ¯ T ( p,s )l , ¯ T ( p,s )r . We present some relevant properties of the Jost solutions to (1.1) in the nexttheorem, which is the analog of Theorem 2.1 that lists the relevant propertiesof the Jost solutions to (1.5).
Theorem 3.6.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, the corresponding Jost solutions to (1.1) satisfy the following: (a) For each n ∈ Z the quantities z − n ψ ( q,r ) n , z n φ ( q,r ) n , z n ¯ ψ ( q,r ) n , z − n ¯ φ ( q,r ) n areeven in z in their respective domains. The quantity z − n ψ ( q,r ) n is analytic in | z | < and continuous in | z | ≤ . (c) The quantity z n φ ( q,r ) n is analytic in | z | < and continuous in | z | ≤ . (d) The quantity z n ¯ ψ ( q,r ) n is analytic in | z | > and continuous in | z | ≥ . (e) The quantity z − n ¯ φ ( q,r ) n is analytic in | z | > and continuous in | z | ≥ . (f) The Jost solution ψ ( q,r ) n has the expansion ψ q,r ) n = ∞ X l = n K ( q,r ) nl z l , | z | ≤ , (3.32) with the double-indexed quantities K ( q,r ) nl for which we have K ( q,r ) nn = D ( q,r ) ∞ − q n D ( q,r ) n D ( q,r ) n − , (3.33) K ( q,r ) n ( n +2) = D ( q,r ) ∞ q n D ( q,r ) n − q n +1 D ( q,r ) n +1 − q n (cid:16) S ( q,r ) ∞ − S ( q,r ) n (cid:17) D ( q,r ) n S ( q,r ) n D ( q,r ) n − , with D ( q,r ) n , D ( q,r ) ∞ , S ( q,r ) n , S ( q,r ) ∞ being the scalar quantities defined in (2.50) , (2.52) , (2.54) , respectively, and that K ( q,r ) nl = 0 when n + l is oddor l < n . (g) The Jost solution ¯ ψ ( q,r ) n has the expansion ¯ ψ q,r ) n = ∞ X l = n ¯ K ( q,r ) nl z l , | z | ≥ , (3.34) with the double-indexed quantities ¯ K ( q,r ) nl for which we have ¯ K ( q,r ) nn = E ( q,r ) ∞ E ( q,r ) n − " r n , (3.35)¯ K ( q,r ) n ( n +2) = E ( q,r ) ∞ − q n r n +1 E ( q,r ) n + Q ( q,r ) ∞ − Q ( q,r ) n − E ( q,r ) n − r n +1 (1 − q n r n ) E ( q,r ) n + r n (cid:16) Q ( q,r ) ∞ − Q ( q,r ) n − (cid:17) E ( q,r ) n − , ith E ( q,r ) n , E ( q,r ) ∞ , Q ( q,r ) n , Q ( q,r ) ∞ being the scalar quantities defined in (2.51) , (2.53) , (2.55) , respectively, and that ¯ K ( q,r ) nl = 0 when n + l is oddor l < n . (h) For the Jost solution φ ( q,r ) n we have the expansion z n φ ( q,r ) n = ∞ X l =0 P ( q,r ) nl z l , | z | ≤ , with the double-indexed quantities P ( q,r ) nl for which we have P ( q,r ) n = D ( q,r ) n − " ,P ( q,r ) n = D ( q,r ) n − " q n − r n − + (1 − q n − r n − ) S ( q,r ) n − − r n − , and that P ( q,r ) nl = 0 when l is odd or l < . (i) For the Jost solution ¯ φ ( q,r ) n we have the expansion z − n ¯ φ ( q,r ) n = ∞ X l =0 ¯ P ( q,r ) nl z l , | z | ≥ , with the double-indexed quantities ¯ P ( q,r ) nl for which we have ¯ P ( q,r ) n = E ( q,r ) n − " − q n − , ¯ P ( q,r ) n = E ( q,r ) n − q n − − q n − q n − r n − − q n − Q ( q,r ) n − Q ( q,r ) n − , and that ¯ P ( q,r ) nl = 0 when l is odd or l < . (j) The scattering coefficients for (1.1) are even in z in their respective do-mains. The domain for the reflection coefficients is the unit circle T andthe domains for the transmission coefficients consist the union of T andtheir regions of extensions. (k) The quantity /T ( q,r ) has an extension from z ∈ T to | z | < and thatextension is analytic for | z | < and continuous for | z | ≤ . Similarly,the quantity / ¯ T ( q,r ) has an extension from z ∈ T so that it is analytic in | z | > and continuous in | z | ≥ . roof. The proof is similar to the proof of Theorem 2.1 and is obtained withthe help of (1.1) and (2.2)–(2.5).In the next theorem, at z = 1 we present the values of the Jost solutionsto (1.1), (1.5), (1.6) when the corresponding potential pairs ( q, r ) , ( u, v ) , ( p, s )are related to each other as in (3.1)–(3.4). These results will be useful in thesolution to the inverse problem for (1.1). Theorem 3.7.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Then, at z = 1 the Jost solutions ψ ( q,r ) n (1) , ¯ ψ ( q,r ) n (1) , ψ ( u,v ) n (1) , ¯ ψ ( u,v ) n (1) , ψ ( p,s ) n (1) , and ¯ ψ ( p,s ) n (1) have the values h ¯ ψ ( q,r ) n (1) ψ ( q,r ) n (1) i = ∞ X j = n r j , (3.36) h ¯ ψ ( u,v ) n (1) ψ ( u,v ) n (1) i = E ( q,r ) n − E ( q,r ) ∞ E ( q,r ) n − D ( q,r ) ∞ ∞ X j = n q j − r n D ( q,r ) n − E ( q,r ) ∞ D ( q,r ) n − D ( q,r ) ∞ − r n ∞ X j = n q j , (3.37) h ¯ ψ ( p,s ) n (1) ψ ( p,s ) n (1) i = E ( q,r ) n − E ( q,r ) ∞ q n ∞ X j = n +1 r j q n E ( q,r ) n − D ( q,r ) ∞ D ( q,r ) n E ( q,r ) ∞ ∞ X j = n +1 r j D ( q,r ) n D ( q,r ) ∞ , (3.38) where D ( q,r ) n , D ( q,r ) ∞ , E ( q,r ) n , E ( q,r ) ∞ , are the quantities defined in (2.50) and (2.51) ,respectively.Proof. One can obtain (3.36) via iteration by directly solving (1.1) with z = 1and using (2.2) and (2.4). Similarly, one can get (3.37) via iteration by solving(1.5) with z = 1 and using (2.2), (2.4), (2.50), (2.51), (3.1), and (3.2). One canobtain (3.38) in a similar manner. Alternatively, one can directly verify that thetwo columns of (3.36) satisfy (1.1) with z = 1 with the respective asymptoticsin (2.4) and (2.2). Similarly, with the help of (2.50), (2.51), (3.1), (3.2), onecan directly verify that the two columns given in (3.37) have the respectiveasymptotics in (2.4) and (2.2) and that they also satisfy (1.5) with z = 1 . Ina similar way, with the help of (2.50), (2.51), (3.3), and (3.4), one can directlyverify that the two columns given in (3.38) have the respective asymptotics in(2.4) and (2.2) and that they each satisfy (1.6) with z = 1 .
29e see that at z = 1 the Jost solutions appearing on the left-hand sidesof (3.36), (3.37), (3.38) can be expressed by using (2.12), (2.14), (2.19), (2.21),(3.32), (3.34) as h ¯ ψ ( q,r ) n (1) ψ ( q,r ) n (1) i = " ∞ X l = n ¯ K ( q,r ) nl ∞ X l = n K ( q,r ) nl , (3.39) h ¯ ψ ( u,v ) n (1) ψ ( u,v ) n (1) i = " ∞ X l = n ¯ K ( u,v ) nl ∞ X l = n K ( u,v ) nl , (3.40) h ¯ ψ ( p,s ) n (1) ψ ( p,s ) n (1) i = " ∞ X l = n ¯ K ( p,s ) nl ∞ X l = n K ( p,s ) nl . (3.41)For a column vector K with two components let use [ K ] and [ K ] to denotethe first and second components, respectively, i.e. we let (cid:2) K (cid:3) := (cid:2) (cid:3) K , (cid:2) K (cid:3) := (cid:2) (cid:3) K . (3.42)In the next theorem we show how to recover the potentials q n and r n from (3.40)and (3.41), respectively. Theorem 3.8.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Then, q n and r n are related tothe Jost solutions for ( u, v ) evaluated at z = 1 given in (3.40) as q n = D ( q,r ) ∞ E ( q,r ) ∞ h ψ ( u,v ) n (1) i h ¯ ψ ( u,v ) n (1) i − h ψ ( u,v ) n +1 (1) i h ¯ ψ ( u,v ) n +1 (1) i , (3.43) r n = − E ( q,r ) ∞ D ( q,r ) ∞ h ¯ ψ ( u,v ) n (1) i h ¯ ψ ( u,v ) n (1) i h ¯ ψ ( u,v ) n (1) i h ψ ( u,v ) n (1) i − h ¯ ψ ( u,v ) n (1) i h ψ ( u,v ) n (1) i . (3.44) Similarly, q n and r n are related to the Jost solutions for ( p, s ) evaluated at z = 1 given in (3.41) as q n = D ( q,r ) ∞ E ( q,r ) ∞ h ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i h ¯ ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i − h ¯ ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i , (3.45) r n = E ( q,r ) ∞ D ( q,r ) ∞ h ¯ ψ ( p,s ) n − (1) i h ψ ( p,s ) n − (1) i − h ¯ ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i . (3.46)30 roof. From (3.37) we obtain D ( q,r ) ∞ E ( q,r ) ∞ h ψ ( u,v ) n (1) i h ¯ ψ ( u,v ) n (1) i = ∞ X j = n q j , (3.47)which yields (3.43). Then, using (3.47) in (3.37) we get (3.44). Similarly, from(3.38) we get E ( q,r ) ∞ D ( q,r ) ∞ h ¯ ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i = ∞ X j = n +1 r j , (3.48)which yields (3.46). Using (3.48) in (3.38) we get (3.45).Let us remark that, as seen from (3.36), we cannot determine q n from eitherside of (3.39) even though we obtain r n as r n = h ¯ ψ ( q,r ) n (1) i − h ¯ ψ ( q,r ) n +1 (1) i . In this section we analyze the bound states for each of the three linear systems(1.1), (1.5), (1.6), and we describe their bound-state data sets in terms of thebound-state z -values, the multiplicity of each bound state, and the bound-statenorming constants. We show how the bound-state norming constants are relatedto the dependency constants and the transmission coefficients. Using a pairof constant matrix triplets ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) , we describe in an elegantmanner each bound-state data set for any number of bound states with anymultiplicities. In the formulation of the Marchenko method, we show how torelate the two matrix triplets to the relevant Marchenko kernels in such a waythat the procedure is generally applicable for both continuous and discrete linearsystems. When the potential pairs ( q, r ) , ( u, v ) , and ( p, s ) are related to eachother as in (3.1)–(3.4), we describe how the corresponding bound-state data setsare related to each other and also how the corresponding pairs of matrix tripletsare related to each other.Let us first consider the bound states for (1.1). By definition a bound statefor (1.1) corresponds to a square-summable solution in n ∈ Z , i.e. a solution (cid:20) α n β n (cid:21) satisfying ∞ X n = −∞ (cid:0) | α n | + | β n | (cid:1) < + ∞ . (4.1)The bound states for (1.5) and (1.6) are defined in a similar way, i.e. for eachof these two systems a bound state corresponds to a square-summable solution.Let us introduce the dependency constants related to bound states for eachof (1.1), (1.5), (1.6). For each of these systems, at a bound state at z = z j φ n and ψ n are linearly dependent because φ n ( z j ) decayssufficiently fast as n → −∞ and ψ n ( z j ) decays sufficiently fast as n → + ∞ so that each of these solutions satisfy (4.1). Thus, a bound-state solution isa constant multiple of either of φ n ( z j ) and ψ n ( z j ) , and we can introduce thedouble-indexed dependency constant γ j as the constant satisfying φ n ( z j ) = γ j ψ n ( z j ) , n ∈ Z . (4.2)As seen from any of the first equalities in (2.60), (2.61), and (2.62), we observethat (4.2) is equivalent to the vanishing of the Wronskian determinant at z = z j for all n ∈ Z , i.e. (cid:12)(cid:12) φ n ( z j ) ψ n ( z j ) (cid:12)(cid:12) = 0 , n ∈ Z , which is also equivalent to the linear dependence of the Jost solutions φ n and ψ n at z = z j . Similarly, at a bound state at z = ¯ z j , the Jost solutions ¯ φ n and ¯ ψ n arelinearly dependent and for any of the systems (1.1), (1.5), (1.6), this can beexpressed in some equivalent forms such as¯ φ n (¯ z j ) = ¯ γ j ¯ ψ n (¯ z j ) , n ∈ Z , where ¯ γ j is the double-indexed dependency constant, and also as (cid:12)(cid:12) ¯ φ n (¯ z j ) ¯ ψ n (¯ z j ) (cid:12)(cid:12) = 0 , n ∈ Z , indicating the linear dependence of the Jost solutions ¯ φ n and ¯ ψ n at z = ¯ z j . If a bound state is not simple, as seen from (2.64), (2.65), (2.67), (2.68),(2.77), and (2.78), the number of constraints is equivalent to the multiplicity ofthe bound state, yielding as many dependency constants as the multiplicity ofthe bound state. At a bound state at z = z j with multiplicity m j , by proceedingas in [7, 10], it follows that each of (2.64), (2.67), (2.77) is equivalent to having m j constraints relating the Jost solutions φ n and ψ n and their z -derivatives as d k φ n ( z j ) dz k = k X l =0 (cid:18) kl (cid:19) γ j ( k − l ) d l ψ n ( z j ) dz l , ≤ k ≤ m j − , (4.3)where (cid:0) kl (cid:1) is the binomial coefficient and we refer to the double-indexed scalarquantities γ jk as the dependency constants at z = z j . Note that (4.3) holdsfor each of the systems (1.1), (1.5), and (1.6). We can use the appropriatesuperscripts so that γ ( q,r ) jk , γ ( u,v ) jk , γ ( p,s ) jk denote the corresponding dependencyconstants for (1.1), (1.5), (1.6), respectively. In a similar way, we obtain thedouble-indexed dependency constants ¯ γ jk at a bound state at z = ¯ z j withmultiplicity ¯ m j , which relate the Jost solutions ¯ φ n and ¯ ψ n and their z -derivativesas d k ¯ φ n (¯ z j ) dz k = k X l =0 (cid:18) kl (cid:19) ¯ γ j ( k − l ) d l ¯ ψ n (¯ z j ) dz l , ≤ k ≤ ¯ m j − . (4.4)32et us also introduce the “residues” t jk of the right transmission coefficientsfor each of (1.1), (1.5), and (1.6) when the corresponding right transmissioncoefficient T r has a pole at z = z j of order m j . Using the expansion T r = t jm j ( z − z j ) m j + t j ( m j − ( z − z j ) m j − + · · · + t j ( z − z j ) + O (1) , z → z j , (4.5)we uniquely obtain the residues t jk for 1 ≤ k ≤ m j and 1 ≤ j ≤ N . We remarkthat t ( q,r ) jk , t ( u,v ) jk , t ( p,s ) jk are defined as in (4.5) by using the right transmissioncoefficients T ( q,r ) , T ( u,v )r , T ( p,s )r corresponding to (1.1), (1.5), (1.6), respectively.In a similar way we define the “residues” ¯ t jk by letting¯ T r = ¯ t j ¯ m j ( z − ¯ z j ) ¯ m j + ¯ t j ( ¯ m j − ( z − ¯ z j ) ¯ m j − + · · · + ¯ t j ( z − ¯ z j ) + O (1) , z → ¯ z j . (4.6)Again using (4.6) with ¯ T ( q,r ) , ¯ T ( u,v )r , ¯ T ( p,s )r we obtain the residues ¯ t ( q,r ) jk , ¯ t ( u,v ) jk , ¯ t ( p,s ) jk corresponding to (1.1), (1.5), (1.6), respectively.In the next theorem we elaborate on the bound states for (1.1). Theorem 4.1.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, we have the following: (a) A bound state for (1.1) can only occur at a z -value for which T ( q,r ) has apole in the region < | z | < or ¯ T ( q,r ) has a pole in the region | z | > . (b) The number of bound states is finite, i.e. the number of poles of T ( q,r ) in < | z | < and the number of poles of ¯ T ( q,r ) in | z | > each must befinite. (c) A bound state is not necessarily simple, but its multiplicity must be finite. (d)
Since each of the transmission coefficients T ( q,r ) and ¯ T ( q,r ) are even in z in their respective domains, the bound-state z -values are symmetricallylocated with respect to the origin of the complex z -plane. (e) At a bound state corresponding to a pole at z j for T ( q,r ) with multiplicity m j , we have the two vectors φ ( q,r ) n ( z j ) d φ ( q,r ) n ( z j ) dzd φ ( q,r ) n ( z j ) dz ... d m j − φ ( q,r ) n ( z j ) dz m j − , ψ ( q,r ) n ( z j ) d ψ ( q,r ) n ( z j ) dzd ψ ( q,r ) n ( z j ) dz ... d m j − ψ ( q,r ) n ( z j ) dz m j − , (4.7)33 elated to each other as in (4.3) via m j dependency constants γ ( q,r ) jk . Sim-ilarly, at the bound state at z = ¯ z j corresponding to a pole of ¯ T ( q,r ) in | z | > , we have the two vectors ¯ φ ( q,r ) n (¯ z j ) d ¯ φ ( q,r ) n (¯ z j ) dzd ¯ φ ( q,r ) n (¯ z j ) dz ... d ¯ m j − ¯ φ ( q,r ) n (¯ z j ) dz ¯ m j − , ¯ ψ ( q,r ) n (¯ z j ) d ¯ ψ ( q,r ) n (¯ z j ) dzd ¯ ψ ( q,r ) n (¯ z j ) dz ... d ¯ m j − ¯ ψ ( q,r ) n (¯ z j ) dz ¯ m j − , (4.8) related to each other as in (4.4) via ¯ m j dependency constants ¯ γ ( q,r ) jk . We re-call that an overbar does not denote complex conjugation and that ψ ( q,r ) n ( z ) ,φ ( q,r ) n ( z ) , ¯ ψ ( q,r ) n ( z ) , ¯ φ ( q,r ) n ( z ) are the four Jost solutions to (1.1) .Proof. By Theorem 3.6 we know that z − n ψ ( q,r ) n ( z ) and z n φ ( q,r ) n ( z ) have analyticextensions from z ∈ T to | z | < . Since a bound-state solution to (1.1) mustsatisfy (4.1), with the help of the first equality in (2.60) and (2.64) we prove thatthe bound states located in | z | < T ( q,r ) . By Theorem 3.6(j) we know that T ( q,r ) contains z as z , and hence the boundstates occur at the poles of T ( q,r ) at z = ± z j for 1 ≤ j ≤ N in 0 < | z | < , each having the multiplicity m j . The finiteness of N and of m j is already knownfrom Theorem 2.8(e). In a similar way, with the help of the second equality in(2.60) and (2.65) we show that the bound states of (1.1) in | z | > z = ± ¯ z j , where the two vectors listed in (4.8) are related to each other as statedin (e) and that ¯ T ( q,r ) has a pole at z = ± ¯ z j with multiplicity ¯ m j . The numberof such ¯ z j -values denoted by ¯ N and each multiplicity ¯ m j are both finite as aconsequence of Theorem 2.8(e).In Theorem 4.1 and its proof, the bound-state z -values and their multiplici-ties are described by the sets {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 without using thesuperscript ( q, r ) . For clarity, one must use z ( q,r ) j , m ( q,r ) j , N ( q,r ) , ¯ z ( q,r ) j , ¯ m ( q,r ) j , ¯ N ( q,r ) for (1.1) and similar notations to describe the bound states for (1.5)and (1.6). Then, the bound states for (1.5) and (1.6) can be described by thecorresponding version of Theorem 4.1.Let us also remark that from (2.23) and the analog of (2.23) for ( p, s ) , weconclude that the bound states for (1.5) and (1.6) can equivalently be describedas in Theorem 4.1 by using either the left transmission coefficients or the righttransmission coefficients. If the three potential pairs ( q, r ) , ( u, v ) , ( p, s ) are34elated to each other as in (3.1)–(3.4) then from Theorem 3.4 it follows that T ( q,r ) = E ( q,r ) ∞ T ( u,v )r = E ( q,r ) ∞ T ( p,s )r = 1 D ( q,r ) ∞ T ( u,v )l = 1 D ( q,r ) ∞ T ( p,s )l , ¯ T ( q,r ) = D ( q,r ) ∞ ¯ T ( u,v )r = D ( q,r ) ∞ ¯ T ( p,s )r = 1 E ( q,r ) ∞ ¯ T ( u,v )l = 1 E ( q,r ) ∞ ¯ T ( p,s )l , (4.9)and hence the sets {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 refer to the common sets ofbound states and the corresponding multiplicities for (1.1), (1.5), (1.6). In thatcase, from (4.9) it follows that the residues corresponding to (1.1), (1.5), (1.6)are related to each other as t ( q,r ) jk = E ( q,r ) ∞ t ( u,v ) jk = E ( q,r ) ∞ t ( p,s ) jk , ¯ t ( q,r ) jk = D ( q,r ) ∞ ¯ t ( u,v ) jk = D ( q,r ) ∞ ¯ t ( p,s ) jk . (4.10)In the next theorem, when the potential pairs ( q, r ) , ( u, v ), ( p, s ) are re-lated to each other as in (3.1)–(3.4), we present the relationships among thecorresponding dependency constants. Theorem 4.2.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Then, the corresponding depen-dency constants γ ( q,r ) jk , γ ( u,v ) jk , γ ( p,s ) jk are related to each other for ≤ k ≤ m j − and ≤ j ≤ N as D ( q,r ) ∞ γ ( q,r ) jk = γ ( p,s ) jk ,D ( q,r ) ∞ γ ( q,r ) jk = k X l =0 (cid:18) kl (cid:19) d l σ ( z j ) dz l γ ( u,v ) j ( k − l ) , (4.11) where we have defined σ ( z ) := 11 − z . (4.12) Similarly, the corresponding dependency constants ¯ γ ( q,r ) jk , ¯ γ ( u,v ) jk , ¯ γ ( p,s ) jk are relatedto each other as ¯ γ ( p,s ) jk = E ( q,r ) ∞ ¯ γ ( q,r ) jk , ¯ γ ( u,v ) jk = E ( q,r ) ∞ k X l =0 (cid:18) kl (cid:19) d l σ (¯ z j ) dz l ¯ γ ( q,r ) j ( k − l ) . (4.13) Proof.
Using (3.16) and (3.18) in the Wronskian determinant on the right-handside of the first equality in (2.60) we get (cid:12)(cid:12)(cid:12) φ ( q,r ) n ψ ( q,r ) n (cid:12)(cid:12)(cid:12) = D ( q,r ) ∞ (cid:16) det h Λ ( q,r ) n i(cid:17) (cid:12)(cid:12)(cid:12) φ ( p,s ) n ψ ( p,s ) n (cid:12)(cid:12)(cid:12) , (4.14)35here Λ ( q,r ) n is the coefficient matrix appearing in (3.16) and (3.18), i.e.Λ ( q,r ) n := E ( q,r ) n − − q n D ( q,r ) n r n E ( q,r ) n − D ( q,r ) n − . (4.15)From (2.50) and (4.15) we see that the determinant of Λ ( q,r ) n is given bydet h Λ ( q,r ) n i = 1 E ( q,r ) n − D ( q,r ) n , (4.16)and hence det[Λ ( q,r ) n ] = 0 for any integer n. Using (4.14), with the help of thefirst equality in (2.62), we obtain a ( q,r ) n ( z ) = D ( q,r ) ∞ (cid:16) det h Λ ( q,r ) n i(cid:17) a ( p,s ) n ( z ) , n ∈ Z . (4.17)From (4.17) we conclude that (2.64) for the potential pair ( q, r ) occurs if andonly if (2.77) for the potential pair ( p, s ) occurs. Comparing (4.3) for ( q, r ) with(4.3) for ( p, s ), with the help of (3.16) and (3.18) and the fact the matrix Λ ( q,r ) n defined in (4.15) is invertible, we establish the equality in the first line of (4.11).In a similar way, with the help of (2.65), (2.78), and (4.4) written for the pairs( q, r ) and ( p, s ) and also using (3.20) and (3.22) we obtain the equality in thefirst line of (4.13). The equality in the second line of (4.11) is established in asimilar manner by using (2.64), (2.67), and (4.3) written for the pairs ( q, r ) and( u, v ) and also using (3.15) and (3.17). The equality in the second line of (4.13)is established in a similar manner by using (2.65), (2.68), and (4.4) written forthe pairs ( q, r ) and ( u, v ) and also using (3.19) and (3.21).As expected, for a unique solution to an inverse problem, for each bound statewe need to specify a corresponding bound-state norming constant. If a boundstate has a multiplicity then we must specify a separate norming constant foreach multiplicity. In the case of (1.1), (1.5), and (1.6), because the bound-statelocations occur symmetrically with respect to the origin of the complex z -plane,we mention that the bound-state norming constants for those symmetric pairscoincide.As a summary, in specifying the bound-state data sets for each of (1.1),(1.5), (1.6), in addition to providing {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 we alsoneed to provide the sets of bound-state norming constants {{ c jk } m j − k =0 } Nj =1 and {{ ¯ c jk } ¯ m j − k =0 } ¯ Nj =1 , where the double-indexed quantities c jk and ¯ c jk denote thenorming constants associated with z j and ¯ z j , respectively. Clearly, we must use c ( q,r ) jk and ¯ c ( q,r ) jk for the norming constants for (1.1), use c ( u,v ) jk and ¯ c ( u,v ) jk for thenorming constants for (1.5), and use c ( p,s ) jk and ¯ c ( p,s ) jk for the norming constantsfor (1.6). In the presence of multiplicities it becomes extremely complicated to36eal with bound states. That is why in the literature most researchers makethe artificial assumption that the bound states are simple.The bound states with multiplicities can easily and in an elegant way behandled [3, 4, 5, 6, 7, 8, 10] for both continuous and discrete systems by usingan appropriate constant matrix triplet ( A, B, C ) for a KdV-like system or apair of triplets (
A, B, C ) and ( ¯ A, ¯ B, ¯ C ) for an NLS-like system. Let us mentionthat the potentials appear in the block-diagonal format in the linear system inthe KdV-like case and the potentials appear in the block off-diagonal format inthe linear system in the NLS-like case. In all these cases, the relevant tool tosolve inverse scattering problems is the Marchenko method. In the continuouscase the Marchenko method involves a linear integral equation known as theMarchenko equation or a system of linear integral equations to which we referas the Marchenko system. In the discrete case the integrals in the Marchenkoequations or in the Marchenko systems are simply replaced by the correspondingsummations. In either the continuous case or the discrete case, the matrix triplet( A, B, C ) in the KdV-like case or the triplets (
A, B, C ) and ( ¯ A, ¯ B, ¯ C ) in the NLS-like case are chosen in such way that the part of the kernel of the Marchenkosystem related to the bound states is expressed in a simple manner in terms ofsuch matrix triplets.In this paper we deal with NLS-like discrete systems, and hence we use thepair of matrix triplets ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ). If there is a bound state at z = z j with multiplicity m j for 1 ≤ j ≤ N, then the triplet ( A, B, C ) can bechosen as A := A · · · A · · · · · · A N , B := B B ... B N , C := (cid:2) C C · · · C N (cid:3) , (4.18)in such a way that A is a block-diagonal matrix, B is a block column vector,and C is a block row vector with A j := z j · · · z j · · · z j · · · · · · z j
10 0 0 . . . z j , B j := , (4.19) C j := (cid:2) c j ( m j − c j ( m j − · · · c j c j (cid:3) . (4.20)As seen from (4.19), A j is an m j × m j matrix in the Jordan canonical formwith z j appearing in the diagonal entries, and B j is an m j × m j −
1) entries and 1 in the m j th entry. As also seen from (4.20) the1 × m j matrix C j is constructed from the norming constants c jk . In our paper,the matrix triplet ( A j , B j , C j ) is chosen to include the contribution from both z = z j and z = − z j , and this will be seen from (4.37) and Theorem 4.7(d).37n a similar way, for the bound states at z = ± ¯ z j with multiplicity ¯ m j for1 ≤ j ≤ ¯ N, the corresponding triplet ( ¯ A, ¯ B, ¯ C ) can be chosen as¯ A := ¯ A · · ·
00 ¯ A · · · · · · ¯ A ¯ N , ¯ B := ¯ B ¯ B ...¯ B ¯ N , ¯ C := (cid:2) ¯ C ¯ C · · · ¯ C ¯ N (cid:3) , (4.21)in such a way that ¯ A is a block-diagonal matrix, ¯ B is a block column vector,and ¯ C is a block row vector with¯ A j := ¯ z j · · · z j · · · z j · · · · · · ¯ z j
10 0 0 · · · z j , ¯ B j := , (4.22)¯ C j := (cid:2) ¯ c j ( ¯ m j − ¯ c j ( ¯ m j − · · · ¯ c j ¯ c j (cid:3) . (4.23)As seen from (4.22), ¯ A j is an ¯ m j × ¯ m j matrix in the Jordan canonical formwith ¯ z j appearing in the diagonal entries, and ¯ B j is an m j × m j − m j th entry. As also seen from (4.23) the1 × ¯ m j matrix ¯ C j is constructed from the norming constants ¯ c jk . In our paper,the matrix triplet ( ¯ A j , ¯ B j , ¯ C j ) is chosen to include the contribution from both z = ¯ z j and z = − ¯ z j , and this will be seen from (4.38) and Theorem 4.7(d).The Marchenko system associated with either of (1.5) and (1.6) is given by (cid:2) ¯ K nm K nm (cid:3) + " n + m Ω n + m + ∞ X l = n +1 (cid:2) ¯ K nl K nl (cid:3) " l + m Ω l + m = " , m > n, (4.24)where we have defined K nm := 12 πi I dz ψ n z − m − , ¯ K nm := 12 πi I dz ¯ ψ n z m − , (4.25) Ω k := ˆ R k + CA k − B, ¯Ω k := ˆ¯ R k + ¯ C ( ¯ A ) − k − ¯ B, k even , Ω k := 0 , ¯Ω k := 0 , k odd , (4.26)with ˆ R k := 12 πi I dz R z k − , ˆ¯ R k := 12 πi I dz ¯ R z − k − . (4.27)We remark that ψ n and ¯ ψ n appearing in (4.25) are the Jost solutions satisfying(2.2) and (2.4), respectively, and that the integral in (4.25) denoted by H is the38ontour integral along the unit circle T in the positive direction. In fact, forthe potential pair ( u, v ) the quantities K nm and ¯ K nm are the column vectorsappearing in (2.12) and (2.14), respectively. The scalar quantities R and ¯ R appearing in (4.27) are the right reflection coefficients, and the matrix triplets( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) appearing in (4.26) are those described in (4.18) and(4.21), respectively.Let us also remark that K nm = 0 and ¯ K nm = 0 when n + m is odd, andthis is already stated in Theorem 2.1, Corollary 2.2, and Theorem 3.6 for thepotential pairs ( u, v ) , ( p, s ) , and ( q, r ) , respectively. Similarly, we already knowthat the scattering coefficients are even in z for each of these three potentialpairs. Hence, from (4.27) we see that ˆ R k = 0 and ˆ¯ R k = 0 when k is odd. Thus,the second line of (4.26) is consistent with (4.24) and (4.27).The derivation of (4.24) is obtained as follows. We can express the Jost so-lutions φ n and ¯ φ n satisfying (2.3) and (2.5), respectively, as linear combinationsof ψ n and ¯ ψ n as φ n T r = ¯ ψ n + ψ n R, ¯ φ n ¯ T r = ψ n + ¯ ψ n ¯ R, (4.28)where T r and ¯ T r are the right transmission coefficients appearing in (2.7) and(2.9), respectively. We use the Fourier transform on (4.28), and for m > n weget 12 πi I dz φ n T r z m − = 12 πi I dz ¯ ψ n z m − + 12 πi I dz ψ n R z m − , (4.29)12 πi I dz ¯ φ n ¯ T r z − m − = 12 πi I dz ψ n z − m − + 12 πi I dz ¯ ψ n ¯ R z − m − , (4.30)yielding the two columns of (4.24).Using the notation of (3.42), from (4.24) we get the two uncoupled scalarequations for m > n as h K nm i + ¯Ω n + m − ∞ X l = n +1 ∞ X j = n +1 h K nj i Ω j + l ¯Ω l + m = 0 , h ¯ K nm i + Ω n + m − ∞ X l = n +1 ∞ X j = n +1 h ¯ K nj i ¯Ω j + l Ω l + m = 0 , (4.31)and once the system (4.31) is solved we also have h ¯ K nm i = − ∞ X l = n +1 h K nl i Ω l + m , h K nm i = − ∞ X l = n +1 h ¯ K nl i ¯Ω l + m . (4.32)39et us recall that K nm = 0 and ¯ K nm = 0 when n + m is odd, and hence thelower indices for the summations in (4.31) and (4.32) actually start with n + 2instead of n + 1 . Nevertheless, we use n + 1 there instead of n + 2 so that(4.31) and (4.32) appear in the standard form as a generic Marchenko systemin the discrete case. When we use (4.31) corresponding to (1.5), we recover thepotentials u n and v n as u n = h K ( u,v ) n ( n +2) i , v n = h ¯ K ( u,v ) n ( n +2) i , (4.33)which are compatible with (2.13) and (2.15), respectively. In the same manner,if we use (4.31) corresponding to (1.6), we recover the potentials p n and s n as p n = h K ( p,s ) n ( n +2) i , s n = h ¯ K ( p,s ) n ( n +2) i , (4.34)which are compatible with (2.20) and (2.22), respectively.Next, we describe the construction of the norming constants c jk and ¯ c jk interms of the residues t jk and ¯ t jk and the dependency constants γ jk and ¯ γ jk . Theorem 4.3.
Assume that the potentials u n and v n appearing in (1.5) arerapidly decaying and − u n v n = 0 for n ∈ Z . Let us use {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 to denote the corresponding sets for the bound-state locations andtheir multiplicities. Then, the norming constants c ( u,v ) jk appearing in (4.20) arerelated to the residues t ( u,v ) jk appearing in (4.5) and the dependency constants γ ( u,v ) jk appearing in (4.3) as c ( u,v ) jk = − m j − − k X l =0 t ( u,v ) j ( k +1+ l ) γ ( u,v ) jl l ! , ≤ j ≤ N, ≤ k ≤ m j − . (4.35) Similarly, the norming constants ¯ c ( u,v ) jk appearing in (4.23) are related to theresidues ¯ t ( u,v ) jk appearing in (4.6) and the dependency constants ¯ γ ( u,v ) jk appearingin (4.4) as ¯ c ( u,v ) jk = 2 ¯ m j − − k X l =0 ¯ t ( u,v ) j ( k +1+ l ) ¯ γ ( u,v ) jl l ! , ≤ j ≤ ¯ N, ≤ k ≤ ¯ m j − . (4.36) Proof.
For notational simplicity, we outline the proof without using the super-script ( u, v ) on the relevant quantities. As seen from (4.26) the contribution tothe Marchenko kernels Ω k and ¯Ω k from the bound states are given by CA k − B and ¯ C ( ¯ A ) − k − ¯ B , respectively. Thus, the contribution to the Marchenko kernelΩ k from the bound state at z = z j is C j A k − j B j / k from the bound state at z = ¯ z j is ¯ C j ( ¯ A j ) − k − ¯ B j / . Thisindicates that the contribution from the pole at z = z j of T r to the left-handside of (4.29) is evaluated as12 πi I dz φ n T r z m − = − ∞ X l = n K nl C j A l + m − j B j . (4.37)40imilarly, the contribution to the left-hand side of (4.30) from the pole at z = ¯ z j of ¯ T r is evaluated as12 πi I dz ¯ φ n ¯ T r z − m − = − ∞ X l = n ¯ K nl ¯ C j ( ¯ A j ) − l − m − ¯ B j . (4.38)Using (4.5) on the left-hand side of (4.37) we evaluate the aforementioned con-tribution as 12 πi I dz φ n T r z m − = m j − X k =0 t jk k ! d k ( φ n z m − ) dz k (cid:12)(cid:12)(cid:12)(cid:12) z = z j . (4.39)Using (4.3) on the right-hand side of (4.39) we write that right-hand side interms of the residues t jk , the dependency constants γ jk , and d k ψ n ( z j ) /dz k . Finally, we write the expansion for d k ψ n ( z j ) /dz k in terms of the double-indexedquantities K nl appearing in (2.12). By equating the result to the right-handside of (4.37), we establish (4.35). We establish (4.36) in a similar manner byevaluating the left-hand side of (4.38) with the help of (4.6) and then by using(4.4) and also by using (2.14).As the next corollary indicates, the result of Theorem 4.3 also holds for thepotential pair ( p, s ) appearing in (1.6). A proof is omitted because it is similarto the proof of Theorem 4.3. Corollary 4.4.
Assume that the potentials p n and s n appearing in (1.6) arerapidly decaying and − p n s n = 0 for n ∈ Z . Let us use {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 to denote the corresponding sets for the bound-state locations andtheir multiplicities. Then, the norming constants c ( p,s ) jk and ¯ c ( p,s ) jk are related tothe corresponding residues t ( p,s ) jk and ¯ t ( p,s ) jk and the dependency constants γ ( p,s ) jk and ¯ γ ( p,s ) jk as c ( p,s ) jk = − m j − − k X l =0 t ( p,s ) j ( k +1+ l ) γ ( p,s ) jl l ! , ≤ j ≤ N, ≤ k ≤ m j − , ¯ c ( p,s ) jk = 2 ¯ m j − − k X l =0 ¯ t ( p,s ) j ( k +1+ l ) ¯ γ ( p,s ) jl l ! , ≤ j ≤ ¯ N , ≤ k ≤ ¯ m j − . (4.40)We note that the norming constants are related to the residues and the de-pendency constants in the same manner both in Theorem 4.3 and Corollary 4.4.Hence, without loss of any generality, for the potential pair ( q, r ) we can definethe norming constants c ( q,r ) jk and ¯ c ( q,r ) jk , the respective row vectors C ( q,r ) j and¯ C ( q,r ) j appearing in (4.20) and (4.23), and the respective row vectors C ( q,r ) and¯ C ( q,r ) appearing in (4.18) and (4.21) in Corollary 4.4. The result is stated next.41 efinition 4.5. Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, the corresponding norming constants c ( q,r ) jk and ¯ c ( q,r ) jk are related to the residues t ( q,r ) jk and ¯ t ( q,r ) jk and the dependencyconstants γ ( q,r ) jk and ¯ γ ( q,r ) jk as c ( q,r ) jk := − m j − − k X l =0 t ( q,r ) j ( k +1+ l ) γ ( q,r ) jl l ! , ≤ j ≤ N, ≤ k ≤ m j − , ¯ c ( q,r ) jk := 2 ¯ m j − − k X l =0 ¯ t ( q,r ) j ( k +1+ l ) ¯ γ ( q,r ) jl l ! , ≤ j ≤ ¯ N , ≤ k ≤ ¯ m j − . (4.41)If the potential pairs ( q, r ) , ( u, v ) , and ( p, s ) are related as in (3.1)–(3.4),then the corresponding residues are related as in (4.10) and the correspondingdependency constants are related as in (4.11) and (4.13). In the next theorem,when (3.1)–(3.4) hold, we show how the corresponding bound-state normingconstants are related to each other. Theorem 4.6.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Then, the corresponding bound-state norming constants c ( u,v ) jk and ¯ c ( u,v ) jk are related to c ( p,s ) jk and ¯ c ( p,s ) jk as C ( u,v ) j = C ( p,s ) j (cid:0) I − A − j (cid:1) , ≤ j ≤ N, ¯ C ( p,s ) j = ¯ C ( u,v ) j (cid:2) I − ( ¯ A j ) − (cid:3) , ≤ j ≤ ¯ N, (4.42) where ( A j , B j , C j ) and ( ¯ A j , ¯ B j , ¯ C j ) are the matrix triplets appearing in (4.19) , (4.20) , (4.22) , and (4.23) . Consequently, we have C ( u,v ) = C ( p,s ) (cid:0) I − A − (cid:1) , ¯ C ( p,s ) = ¯ C ( u,v ) (cid:2) I − ( ¯ A ) − (cid:3) , (4.43) where ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) are the matrix triplets appearing in (4.18) and (4.21) , respectively. Similarly, the norming constants c ( q,r ) jk and ¯ c ( q,r ) jk are relatedto the norming constants c ( p,s ) jk and ¯ c ( p,s ) jk as C ( q,r ) j = E ( q,r ) ∞ D ( q,r ) ∞ C ( p,s ) j , ≤ j ≤ N, ¯ C ( q,r ) j = D ( q,r ) ∞ E ( q,r ) ∞ ¯ C ( p,s ) j , ≤ j ≤ ¯ N, (4.44)42 nd hence we also have C ( p,s ) = D ( q,r ) ∞ E ( q,r ) ∞ C ( q,r ) , ¯ C ( p,s ) = E ( q,r ) ∞ D ( q,r ) ∞ ¯ C ( q,r ) , (4.45) where D ( q,r ) ∞ and E ( q,r ) ∞ are the constants appearing in (2.50) and (2.51) , re-spectively. Consequently, we get C ( u,v ) = D ( q,r ) ∞ E ( q,r ) ∞ C ( q,r ) (cid:0) I − A − (cid:1) , ¯ C ( u,v ) = E ( q,r ) ∞ D ( q,r ) ∞ ¯ C ( q,r ) (cid:2) I − ( ¯ A ) − (cid:3) − . (4.46) Proof.
We will provide only the proof of the first line of (4.42) because thesecond line of (4.42) can be proved in a similar manner. We note that the firstline of (4.10) yields t ( p,s ) jk = t ( u,v ) jk , (4.47)and from (4.11) we have γ ( p,s ) jk = k X l =0 (cid:18) kl (cid:19) d l σ ( z j ) dz l γ ( u,v ) j ( k − l ) , (4.48)where we recall that σ ( z ) is the scalar quantity defined (4.12). For the matrix A j defined in (4.19) we have A − j = z j − z j z j · · · ( − mj ( m j − z mjj ( − mj +1 m j z mj +1 j z j − z j · · · ( − mj − ( m j − z mj − j ( − mj ( m j − z mjj z j · · · ( − mj − ( m j − z mj − j ( − mj − ( m j − z mj − j ... ... ... . . . ... ...0 0 0 · · · z j − z j · · · z j . (4.49)Using (4.12) and (4.47)–(4.49) on the right-hand side of the first line of (4.40),we establish the first line of (4.42). By using the summation over all the boundstates, i.e. summing over 1 ≤ j ≤ N, from the first line of (4.42) we obtainthe first line of (4.43). In a similar manner, the second line of (4.42) yields thesecond line of (4.43). Finally, the proof of (4.44) and (4.45) are obtained byusing (4.10), the first lines of (4.11) and (4.13), and (4.41), and by comparingthe result with (4.40). 43ecall that we use t jk , γ jk , and c jk to denote the residues, the dependencyconstants, and the norming constants, respectively, corresponding to a boundstate at z = z j with multiplicity m j for each of the linear systems (1.1), (1.5),and (1.6). In the next theorem we compare those quantities with the corre-sponding quantities related to the bound state at z = − z j . We also show thatthe contributions to the Marchenko kernels from z = z j and from z = − z j areequal. Theorem 4.7.
For each of the linear systems (1.1) , (1.5) , and (1.6) , as indi-cated in Theorem 4.1, let the bound states and their multiplicities be described bythe sets {± z j , m j } Nj =1 and {± ¯ z j , ¯ m j } ¯ Nj =1 . Let the residues t jk and ¯ t jk be definedas in (4.5) and (4.6) ; the dependency constants γ jk and ¯ γ jk be defined as in (4.3) and (4.4) , respectively; and the norming constants c jk and ¯ c jk be definedas in (4.35) and (4.36) , respectively, or equivalently as in (4.40) or (4.41) . Wehave the following: (a) Let t jk | z = z j and t jk | z = − z j denote the residues at z = z j and z = − z j , respectively. Similarly, let ¯ t jk | z =¯ z j and ¯ t jk | z = − ¯ z j denote the residues at z = ¯ z j and z = − ¯ z j , respectively. We then have t jk (cid:12)(cid:12) z = − z j = ( − k t jk (cid:12)(cid:12) z = z j , ≤ k ≤ m j , ¯ t jk (cid:12)(cid:12) z = − ¯ z j = ( − k ¯ t jk (cid:12)(cid:12) z =¯ z j , ≤ k ≤ ¯ m j . (4.50)(b) Let γ jk | z = z j and γ jk | z = − z j denote the dependency constants at z = z j and z = − z j , respectively. Similarly, let ¯ γ jk | z =¯ z j and ¯ γ jk | z = − ¯ z j denote thedependency constants at z = ¯ z j and z = − ¯ z j , respectively. We then have γ jk (cid:12)(cid:12) z = − z j = ( − k γ jk (cid:12)(cid:12) z = z j , ≤ k ≤ m j − , ¯ γ jk (cid:12)(cid:12) z = − ¯ z j = ( − k ¯ γ jk (cid:12)(cid:12) z =¯ z j , ≤ k ≤ ¯ m j − . (4.51)(c) Let c jk | z = z j and c jk | z = − z j denote the norming constants at z = z j and z = − z j , respectively. Similarly, let ¯ c jk | z =¯ z j and ¯ c jk | z = − ¯ z j denote thenorming constants at z = ¯ z j and z = − ¯ z j , respectively. We then have c jk (cid:12)(cid:12) z = − z j = ( − k c jk (cid:12)(cid:12) z = z j , ≤ k ≤ m j − , ¯ c jk (cid:12)(cid:12) z = − ¯ z j = ( − k ¯ c jk (cid:12)(cid:12) z =¯ z j , ≤ k ≤ ¯ m j − . (4.52)(d) The contribution to the Marchenko kernel Ω n + m from z = z j is equal to thecontribution from z = − z j . Similarly, the contribution to the Marchenkokernel ¯Ω n + m from z = ¯ z j is equal to the contribution from z = − ¯ z j . Proof.
The proof of (a) is obtained as follows. We know that the transmissioncoefficients T r for each of these three linear systems contain z as z . From (4.5),44sing T r ( − z ) = T r ( z ) , as z → − z j we obtain T r ( z ) = ( − m j t jm j ( z + z j ) m j + ( − m j − t j ( m j − ( z + z j ) m j − + · · · + ( − t j ( z + z j ) + O (1) , which yields the first line of (4.50). The second line of (4.50) is obtained from(4.6) by proceeding in a similar manner. This completes the proof of (a). Letus now prove (b). From Theorem 2.1(a) and its analogs in Corollary 2.2 andTheorem 3.6(a), we get ψ n ( − z ) = ( − n ψ n ( z ) , φ n ( − z ) = ( − n φ n ( z ) , ¯ ψ n ( − z ) = ( − n ¯ ψ n ( z ) , ¯ φ n ( − z ) = ( − n ¯ φ n ( z ) . (4.53)Using the first line of (4.53) in (4.3) we determine the dependency constant γ jk | z = − z j and establish the first line of (4.51). The second line of (4.51) isobtained in a similar way by using the second line of (4.53) in (4.4). Thiscompletes the proof of (b). To prove (c) we proceed as follows. Using thefirst lines of (4.50) and (4.51) in (4.35) we determine the norming constant c jk (cid:12)(cid:12) z = − z j and establish the first line of (4.52). The second line of (4.52) isproved in a similar way by using the second lines of (4.50) and (4.51) in (4.36).This completes the proof of (c). Let us finally prove (d). The right-hand sideof (4.37) is the contribution to the Marchenko kernel Ω n + m from the boundstate at z = z j . Using (4.37) and (4.39), with the help of (4.19) and (4.20), weevaluate the contribution to the Marchenko kernel Ω n + m from the bound state z = − z j and we obtain12 πi I dz φ n T r z m − = − ∞ X l = n K nl (cid:0) C j A l + m − j B j (cid:1)(cid:12)(cid:12)(cid:12) z j z j , (4.54)where z j
7→ − z j is used to indicate the substitution of − z j for z j . From (4.20)and the first line of (4.52) we get C j (cid:12)(cid:12) z = − z j = C j (cid:12)(cid:12) z = z j diag (cid:8) ( − m j , ( − m j − , · · · , ( − (cid:9) , (4.55)where diag is used to denote the diagonal matrix. Similarly, from (4.19), forany integer n we get A nj B j (cid:12)(cid:12) z = − z j = diag n ( − n − ( m j − , ( − n − ( m j − , · · · , ( − n o A nj B j (cid:12)(cid:12) z = z j . (4.56)Using (4.55) and (4.56), when n + m and n + l are both even integers in (4.54),we confirm that the right-hand side of (4.54) is equal to the right-hand side of(4.37). Hence the contribution to the Marchenko kernel Ω n + m from z = z j isequal to the contribution from z = − z j . Similarly, we prove that the contributionto ¯Ω n + m from z = − ¯ z j is equal to the contribution from z = ¯ z j . c ( u,v ) jk from the first line of (4.42)we postmultiply that first line by a column vector with m j components so thatwe get c ( u,v ) jk = C ( p,s ) j (cid:0) I − A − j (cid:1) e m j − − k , ≤ k ≤ m j − , where we use e l for the column vector with m j components with all the entries0 except 1 in the l th entry. In a similar way, from the second line of (4.42) weobtain ¯ c ( p,s ) jk = ¯ C ( u,v ) j (cid:2) I − ( ¯ A j ) − (cid:3) ¯ e ¯ m j − − k , ≤ k ≤ ¯ m j − , where we use ¯ e l for the column vector with ¯ m j components having 1 at the l thentry and 0 elsewhere.Let us also remark that the fundamental result given in (4.43) is compatiblewith (4.26), from which we obtain the Marchenko kernels Ω ( u,v ) k and ¯Ω ( u,v ) k andthe Marchenko kernels Ω ( p,s ) k and ¯Ω ( p,s ) k . When (3.1)–(3.4) hold, we see that(3.28) and (3.29), respectively, yield R ( u,v ) = (cid:18) − z (cid:19) R ( p,s ) , ¯ R ( p,s ) = (cid:18) − z (cid:19) ¯ R ( u,v ) . (4.57)Using (4.57) in (4.27) we obtain ˆ R ( u,v ) k = ˆ R ( p,s ) k − ˆ R ( p,s ) k − , ˆ¯ R ( p,s ) k = ˆ¯ R ( u,v ) k − ˆ¯ R ( p,s ) k +2 . (4.58)From (4.26) and (4.58) we see that in the absence of bound states we have Ω ( u,v ) k = Ω ( p,s ) k − Ω ( p,s ) k − , ¯Ω ( p,s ) k = ¯Ω ( u,v ) k − ¯Ω ( u,v ) k +2 . (4.59)In fact, (4.59) holds even in the presence of bound state. Then, comparing(4.58) and (4.59) we get C ( u,v ) A k − B = C ( p,s ) A k − B − C ( p,s ) A k − B, ¯ C ( p,s ) ( ¯ A ) − k − B = ¯ C ( u,v ) ( ¯ A ) − k − ¯ B − ¯ C ( u,v ) ( ¯ A ) − k − ¯ B, which yield the important result given in (4.43). Let us also mention that from464.59) we get Ω ( p,s ) k = ∞ X l =0 Ω ( u,v ) k − l , ¯Ω ( u,v ) k = ∞ X l =0 ¯Ω ( p,s ) k +2 l . (4.60)In the next theorem we show that, when the potential pairs ( u, v ) and ( p, s )are related to each other as in (3.1)–(3.4), their corresponding Marchenko sys-tems hold simultaneously. Theorem 4.8.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pair ( u, v ) appearing in (1.5) and the potential pair ( p, s ) appearing in (1.6) are related to ( q, r ) as in (3.1) – (3.4) . Then, the Marchenko system related to (1.5) holds ifand only if the Marchenko system related to (1.6) holds.Proof. The proof is lengthy and it involves some fine estimates. Let us define W ( u,v ) nm := K ( u,v ) nm + ∞ X l = n ¯ K ( u,v ) nl ¯Ω ( u,v ) l + m , (4.61)¯ W ( u,v ) nm := ¯ K ( u,v ) nm + ∞ X l = n K ( u,v ) nl Ω ( u,v ) l + m , (4.62) W ( p,s ) nm := K ( p,s ) nm + ∞ X l = n ¯ K ( p,s ) nl ¯Ω ( p,s ) l + m , ¯ W ( p,s ) nm := ¯ K ( p,s ) nm + ∞ X l = n K ( p,s ) nl Ω ( p,s ) l + m . (4.63)As seen from (4.24) and the first two equations in (2.13) and (2.15) we need toprove the equivalence of the Marchenko system ¯ W ( u,v ) nm = 0 , m > n,W ( u,v ) nm = 0 , m > n, (4.64)and the Marchenko system ¯ W ( p,s ) nm = 0 , m > n,W ( p,s ) nm = 0 , m > n. (4.65)We provide the proof by relating ¯ W ( u,v ) nm to ¯ W ( p,s ) nm . The relation between W ( u,v ) nm and W ( p,s ) nm can be established in a similar manner and hence that proof will beomitted. Using (3.15) and (3.16) we relate ψ ( u,v ) n and ψ ( p,s ) n to each other and47pply H dz z − m − / (2 πi ) on the resulting equality. Similarly, using (3.19) and(3.20) we relate ¯ ψ ( u,v ) n and ¯ ψ ( p,s ) n to each other and apply H dz z m − / (2 πi ) onthe resulting equality. Then, with the help of (3.1) and (3.4) we obtain " s n − K ( u,v ) nm − " K ( u,v ) n ( m +2) = " − u n s n − K ( p,s ) nm , " s n − ¯ K ( u,v ) nm = " − u n s n − ¯ K ( p,s ) nm − " ¯ K ( p,s ) n ( m − , where we have also used (4.25) for ( u, v ) and ( p, s ) . Using (4.61) and the firstline of (4.59), after some simplifications, for m > n we write ¯ W ( u,v ) nm appearingin (4.62) as¯ W ( u,v ) nm = " − u n s n − u n s n − ¯ W ( p,s ) nm − " s n − ¯ W ( p,s ) n ( m − . (4.66)From (4.66), when m > n + 2 we conclude that the first line of (4.64) holds ifand only if the first line of (4.65) holds. We must analyze the case m = n + 2separately because of the appearance of ¯ W ( p,s ) n ( m − in (4.66). Toward that goal weapply H dz z n − / (2 πi ) on both sides of the first line of (4.28) with the potentialpair ( p, s ). We then get12 πi I dz φ ( p,s ) n T ( p,s )r z n − = 12 πi I dz ¯ ψ ( p,s ) n z n − + 12 πi I dz ψ ( p,s ) n R ( p,s ) z n − . (4.67)In this case, besides the bound-state poles of T ( p,s )r in 0 < | z | < , also the pointat z = 0 contributes to the integral on left-hand side of (4.67). With the helpof (4.25), (4.27), (4.37), from (4.67) we get h z n φ ( p,s ) n i (cid:12)(cid:12)(cid:12)(cid:12) z =0 T ( p,s )r (0) − ∞ X l = n K ( p,s ) nl C ( p,s ) A l + n − B = ¯ K ( p,s ) nn + ∞ X l = n K ( p,s ) nl ˆ R ( p,s ) l + n . (4.68)Using the analogs of (2.16), (2.17), and (2.38) for the potential pair ( p, s ) , wehave h z n φ ( p,s ) n i (cid:12)(cid:12)(cid:12)(cid:12) z =0 T ( p,s )r (0) = D ( p,s ) ∞ D ( p,s ) n − (cid:20) − s n − (cid:21) . (4.69)With the help of (4.69) and the first equality in (4.26) we write (4.68) as¯ K ( p,s ) nn + ∞ X l = n K ( p,s ) nl Ω ( p,s ) l + n = D ( p,s ) ∞ D ( p,s ) n − (cid:20) − s n − (cid:21) , W ( p,s ) nn = D ( p,s ) ∞ D ( p,s ) n − (cid:20) − s n − (cid:21) . (4.70)Because of (4.70) we see that the second term on the right-hand side of (4.66)vanishes when m = n + 2. Consequently, we conclude that the first lines of(4.64) and (4.65) hold simultaneously also when m = n + 2. In this section, when the potentials q n and r n appearing in (1.1) are rapidly de-caying and satisfy (1.2), we provide the solution to the direct scattering problemfor (1.1), i.e. the determination of the scattering coefficients and the bound-stateinformation for (1.1) when the potential pair ( q, r ) is given. The steps in thesolution to the direct problem are outlined as follows:(a) Using ( q n , r n ) in (1.1), we solve (1.1) with the asymptotic conditions (2.2)–(2.5) and uniquely construct the four Jost solutions ψ ( q,r ) n , φ ( q,r ) n , ¯ ψ ( q,r ) n , ¯ φ ( q,r ) n .(b) We recover the scattering coefficients T ( q,r ) , ¯ T ( q,r ) , R ( q,r ) , ¯ R ( q,r ) , L ( q,r ) , ¯ L ( q,r ) by using the asymptotics in (2.6)–(2.9) of the already constructedfour Jost solutions.(c) Next, we determine the poles and their multiplicities for the transmissioncoefficient T ( q,r ) in 0 < | z | < T ( q,r ) in | z | >
1. Note that such poles occurin pairs. We use the notation that the poles of T ( q,r ) in 0 < | z | < z = ± z j and the multiplicity of the pole at each of z = z j and z = − z j is m j for 1 ≤ j ≤ N . Thus, the set {± z j , m j } Nj =1 is uniquely determinedfrom the poles of T ( q,r ) in 0 < | z | <
1. In a similar way, we use ¯ T ( q,r ) todetermine its poles in | z | > | z | > z = ± ¯ z j for 1 ≤ j ≤ ¯ N andthe multiplicity of the pole at each of z = ¯ z j and z = − ¯ z j is ¯ m j . Thus, wealso obtain the set {± ¯ z j , ¯ m j } ¯ Nj =1 . (d) With the help of (4.5) with T ( q,r ) , we determine the residues t ( q,r ) jk for1 ≤ j ≤ N and 1 ≤ k ≤ m j . Similarly, with the help of (4.6) with ¯ T ( q,r ) , we obtain the residues ¯ t ( q,r ) jk for 1 ≤ j ≤ ¯ N and 1 ≤ k ≤ ¯ m j . (e) Using (4.3) for the potential pair ( q, r ), we determine the dependencyconstants γ ( q,r ) jk for 1 ≤ j ≤ N and 0 ≤ k ≤ m j − . Similarly, using (4.4)with the potential pair ( q, r ) we obtain the dependency constants ¯ γ ( q,r ) jk for 1 ≤ j ≤ ¯ N and 0 ≤ k ≤ ¯ m j − . t ( q,r ) jk and ¯ t ( q,r ) jk and the dependency con-stants γ ( q,r ) jk and ¯ γ ( q,r ) jk , from (4.41) we obtain the bound-state normingconstants c ( q,r ) jk and ¯ c ( q,r ) jk . Note that we also get the triplets (
A, B, C ( q,r ) )and ( ¯ A, ¯ B, ¯ C ( q,r ) ) via (4.18)–(4.23). In this section we introduce the linear system (6.1) resembling (4.24), and wecall it the Marchenko system for (1.1). The system (6.1) uses as input thescalar quantities Ω ( q,r ) k and ¯Ω ( q,r ) k , which are defined as in (4.26), and hence itis appropriate that we refer to (6.1) as the Marchenko system for (1.1). Wealso describe how to obtain the potentials q n and r n from the solution to theMarchenko system (6.1).The formulation of the Marchenko system for (1.1) is a significant step inthe analysis of inverse problems related to integrable evolution equations. Weexpect that our method of formulating (6.1) can be applied on some other linearsystems, both in the continuous and discrete cases, for which a directly relevantMarchenko theory has not yet been established.Even though a Marchenko system such as (6.1) directly related to (1.1) isdesirable, such a system does not yet seem to exist in the literature. We obtain(6.1) by exploiting the connection between (1.1) and (1.6). The only slightdifference from the standard Marchenko theory is that the formulas for q n and r n are expressed in terms of the solution to the Marchenko system (6.1) not asin (4.33) or (4.34) but as in (6.13) and (6.14).In the next theorem we present the derivation of the Marchenko system for(1.1). Theorem 6.1.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, the Marchenko system given in (4.24) holds with the relevant quantities listed in (4.25) – (4.27) all related to (1.1) , i.e.we have h ¯ M ( q,r ) nm M ( q,r ) nm i + ( q,r ) n + m Ω ( q,r ) n + m + ∞ X l = n +1 h ¯ M ( q,r ) nl M ( q,r ) nl i ( q,r ) l + m Ω ( q,r ) l + m = " , m > n. (6.1) Here, the scalar quantities Ω ( q,r ) k and ¯Ω ( q,r ) k are related to the scattering data set or (1.1) as in (4.26) , i.e. Ω ( q,r ) k := ˆ R ( q,r ) k + C ( q,r ) A k − B, k even , ¯Ω ( q,r ) k := ˆ¯ R ( q,r ) k + ¯ C ( q,r ) ( ¯ A ) − k − ¯ B, k even , Ω ( q,r ) k := 0 , ¯Ω ( q,r ) k := 0 , k odd , (6.2) with ˆ R ( q,r ) k and ˆ¯ R ( q,r ) k being related to the reflection coefficients R ( q,r ) and ¯ R ( q,r ) as in (4.27) and the matrix triplets ( A, B, C ( q,r ) ) and ( ¯ A, ¯ B, ¯ C ( q,r ) ) are as in (4.18) and (4.21) , respectively.Proof. A direct proof can be given by using the procedure described in (4.28)–(4.30). We present an alternate proof, and this is done by exploiting the con-nection between (1.1) and (1.6) when the potential pairs ( q, r ) and ( p, s ) arerelated as in (3.3) and (3.4). Starting with the Marchenko system (4.24) withthe relevant quantities all related to the potential pair ( p, s ) of (1.6), we trans-form that Marchenko system and the relevant quantities so that they are allrelated to the potential pair ( q, r ) of (1.1). From (3.16) and (3.20) we see that ψ ( p,s ) n = 1 D ( q,r ) ∞ (cid:16) Λ ( q,r ) n (cid:17) − ψ ( q,r ) n , ¯ ψ ( p,s ) n = 1 E ( q,r ) ∞ (cid:16) Λ ( q,r ) n (cid:17) − ¯ ψ ( q,r ) n , (6.3)where Λ ( q,r ) n is the matrix defined in (4.15) and the quantities D ( q,r ) ∞ and E ( q,r ) ∞ are the scalar constants appearing in (2.50) and (2.51), respectively. From (4.16)we know that the matrix Λ ( q,r ) n is invertible for all n ∈ Z . Thus, with the helpof (4.25) and (6.3) we conclude that K ( p,s ) nm = 1 D ( q,r ) ∞ (cid:16) Λ ( q,r ) n (cid:17) − K ( q,r ) nm , ¯ K ( p,s ) nm = 1 E ( q,r ) ∞ (cid:16) Λ ( q,r ) n (cid:17) − ¯ K ( q,r ) nm . (6.4)From the second equalities in (3.28) and (3.29), with the help of (4.27) we obtainˆ R ( p,s ) = D ( q,r ) ∞ E ( q,r ) ∞ ˆ R ( q,r ) , ˆ¯ R ( p,s ) = E ( q,r ) ∞ D ( q,r ) ∞ ˆ¯ R ( q,r ) . (6.5)From the second equality in (3.25) it follows that the poles of T ( p,s )r and T ( q,r ) coincide, and from the second equality in (3.27) we see that the poles of ¯ T ( p,s )r and ¯ T ( q,r ) coincide. Hence, the matrices A, ¯ A, B, ¯ B appearing in (4.26) arecommon to the potential pairs ( p, s ) and ( q, r ) . Using (4.45) and (6.5) in (4.26)we conclude thatΩ ( p,s ) k = D ( q,r ) ∞ E ( q,r ) ∞ Ω ( q,r ) k , ¯Ω ( p,s ) k = E ( q,r ) ∞ D ( q,r ) ∞ ¯Ω ( q,r ) k . (6.6)With the help of the first equalities in (2.20) and (2.22), we observe that theMarchenko system (4.24) related to the potential pair ( p, s ) is equivalent to the51ystem given in (4.65). Using (6.4) and (6.6) we transform (4.65) into ¯ K ( q,r ) nm + ∞ X l = n K ( q,r ) nl Ω ( q,r ) l + m = 0 , m > n,K ( q,r ) nm + ∞ X l = n ¯ K ( q,r ) nl ¯Ω ( q,r ) l + m = 0 , m > n. (6.7)The system in (6.7) can be written in the matrix form as h ¯ K ( q,r ) nm K ( q,r ) nm i + h ¯ K ( q,r ) nn K ( q,r ) nn i ( q,r ) n + m Ω ( q,r ) n + m + ∞ X l = n +1 h ¯ K ( q,r ) nl K ( q,r ) nl i ( q,r ) l + m Ω ( q,r ) l + m = " , m > n. (6.8)The matrix [ ¯ K ( p,s ) nn K ( p,s ) nn ] , as seen from the first equalities in (2.20) and (2.22),is equal to the 2 × K ( p,s ) nn K ( p,s ) nn ] . However, the matrix[ ¯ K ( q,r ) nn K ( q,r ) nn ] appearing in (6.8) is not equal to the identity matrix. From(3.33) and (3.35) it follows that h ¯ K ( q,r ) nn K ( q,r ) nn i = Λ ( q,r ) n E ( q,r ) ∞ D ( q,r ) ∞ , (6.9)where we recall that Λ ( q,r ) n is the invertible matrix appearing in (4.15). Hence,the matrix on the left-hand side of (6.9) is invertible and we have h ¯ K ( q,r ) nn K ( q,r ) nn i − = E ( q,r ) ∞
00 1 D ( q,r ) ∞ (cid:16) Λ ( q,r ) n (cid:17) − . (6.10)Premultiplying both sides of (6.8) by [ ¯ K ( q,r ) nn K ( q,r ) nn ] − , we obtain (6.1), wherewe have defined h ¯ M ( q,r ) nm M ( q,r ) nm i := h ¯ K ( q,r ) nn K ( q,r ) nn i − h ¯ K ( q,r ) nm K ( q,r ) nm i . (6.11)Note that (6.11) implies that M ( q,r ) nm = 0 and ¯ M ( q,r ) nm = 0 when n + m is oddbecause we have K ( q,r ) nm = 0 and ¯ K ( q,r ) nm = 0 when n + m is odd as stated inTheorem 3.6.We can uncouple the Marchenko system (6.1) as in (4.31) and (4.32). Hence,without a proof we state the result in the next corollary.52 orollary 6.2. Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, the Marchenko system (6.1) is equiv-alent to the uncoupled system, for m > n , given by h M ( q,r ) nm i + ¯Ω ( q,r ) n + m − ∞ X l = n +1 ∞ X j = n +1 h M ( q,r ) nj i Ω ( q,r ) j + l ¯Ω ( q,r ) l + m = 0 , h ¯ M ( q,r ) nm i + Ω ( q,r ) n + m − ∞ X l = n +1 ∞ X j = n +1 h ¯ M ( q,r ) nj i ¯Ω ( q,r ) j + l Ω ( q,r ) l + m = 0 , (6.12) and with [ ¯ M ( q,r ) nm ] and [ M ( q,r ) nm ] obtained from the solution to (6.12) as h ¯ M ( q,r ) nm i = − ∞ X l = n +1 h M ( q,r ) nl i Ω ( q,r ) l + m , h M ( q,r ) nm i = − ∞ X l = n +1 h ¯ M ( q,r ) nl i ¯Ω ( q,r ) l + m , where we recall that [ · ] and [ · ] denote the first and second components of therelevant column vectors, as indicated in (3.42) . In the next theorem we describe the recovery of q n and r n from the solutionto the Marchenko system (6.1). Theorem 6.3.
Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Then, q n and r n are recovered from thesolution to the Marchenko system given in (6.1) via q n = ∞ X l = n h M ( q,r ) nl i ∞ X k = n h M ( q,r ) nk i ∞ X l = n h ¯ M ( q,r ) nl i ∞ X k = n h M ( q,r ) nk i − ∞ X l = n h M ( q,r ) nl i ∞ X k = n h ¯ M ( q,r ) nk i , (6.13) r n = ∞ X l = n − h ¯ M ( q,r )( n − l i ∞ X l = n − h M ( q,r )( n − l i − ∞ X l = n h ¯ M ( q,r ) nl i ∞ X l = n h M ( q,r ) nl i , (6.14) where [ · ] and [ · ] denote the first and second components of the relevant columnvectors, as indicated in (3.42) .Proof. Using (3.16), (3.20), and (4.25) we get h ¯ K ( q,r ) nl K ( q,r ) nl i = Λ ( q,r ) n h ¯ K ( p,s ) nl K ( p,s ) nl i E ( q,r ) ∞ D ( q,r ) ∞ , (6.15)53here Λ ( q,r ) n is the invertible matrix in (4.15); D ( q,r ) ∞ and E ( q,r ) ∞ are the scalarconstants appearing in (2.50) and (2.51), respectively; K ( q,r ) nl and ¯ K ( q,r ) nl arethe column vectors in (3.32) and (3.34), respectively; K ( p,s ) nl and ¯ K ( p,s ) nl are thecolumn vectors in (2.19) and (2.21), respectively. With the help of (6.10), from(6.15) for l ≥ n we get h ¯ K ( q,r ) nn K ( q,r ) nn i − h ¯ K ( q,r ) nl K ( q,r ) nl i = E ( q,r ) ∞ D ( q,r ) ∞ − h ¯ K ( p,s ) nl K ( p,s ) nl i E ( q,r ) ∞ D ( q,r ) ∞ . (6.16)We note that the left-hand side of (6.16) is equal to the left-hand side of (6.11).Using the summation with l ≥ n, from (6.16) we obtain ∞ X l = n h ¯ M ( q,r ) nl M ( q,r ) nl i = E ( q,r ) ∞ D ( q,r ) ∞ − ∞ X l = n h ¯ K ( p,s ) nl K ( p,s ) nl i E ( q,r ) ∞ D ( q,r ) ∞ . (6.17)We remark that the summation on the right-hand side in (6.17) is related to[ ¯ ψ ( p,s ) n ψ ( p,s ) n ] evaluated at z = 1, as seen from (2.19) and (2.21). With thehelp of (3.38) and (3.41) we express the right-hand side of (6.17) in terms ofthe matrix on the right-hand side of (3.38), and we get ∞ X l = n h ¯ M ( q,r ) nl M ( q,r ) nl i = E ( q,r ) n − E ( q,r ) ∞ q n ∞ X j = n +1 r j q n E ( q,r ) n − E ( q,r ) ∞ D ( q,r ) n D ( q,r ) ∞ ∞ X j = n +1 r j D ( q,r ) n D ( q,r ) ∞ . (6.18)Using the notation of (3.42), from the (2 ,
1) and (2 ,
2) entries in (6.18) we obtain(6.14). Then, from the (1 ,
1) and (1 ,
2) entries in (6.18) and using (6.14), weobtain (6.13).
In this section we derive the pair of scalar Marchenko equations given in (7.7)and (7.8), which resembles the uncoupled Marchenko system given in (4.31).We refer to the uncoupled system composed of (7.7) and (7.8) as the alternateMarchenko system. Such a system is the discrete analog of the Marchenkosystem given in (6 .
22) and (6 .
23) of [5] in the continuous case. In this section54e also show that the potentials q n and r n are recovered as in (7.5) and (7.6)from the solution to the alternate Marchenko system.We remark that the uncoupled alternate Marchenko equation (7.7) is closelyrelated to the system (1.5) with the potential pair ( u, v ) , and hence we use thesuperscript ( u, v ) in the quantities appearing in (7.7). Similarly, the uncoupledalternate Marchenko equation (7.8) involves the quantities closely related to(1.6) with the potential pair ( p, s ) , and hence we use the superscript ( p, s ) inthe quantities appearing in (7.8). Our alternate Marchenko equations (7.7) and(7.8) and our recovery formulas (7.5) and (7.6) are closely related to (4.12c),(4.12d), (4.21a), and (4.21b), respectively, of [13]. We remark that Tsuchida in[13] assumes that the bound states are all simple, and we also mention that,contrary to our own (7.7) and (7.8), Tsuchida’s (4.12c) and (4.12d) in [13] lackthe appropriate symmetry for a standard Marchenko system apparent in (4.31)in the discrete case.Let us make a comparison between the alternate Marchenko system used inthis section and the Marchenko system introduced in Section 6. The Marchenkosystem (6.1) has the same standard form used in other inverse problems arisingin applications, but the recovery of the potentials q n and r n from the solutionto (6.1) is not standard, i.e. the recovery is not of the form given in (4.33) or(4.34). On the other hand, certain terms in the alternate Marchenko systeminvolve some discrete spacial derivatives and hence the alternate Marchenkosystem slightly differs from the standard Marchenko system (4.24). However,the recovery of the potentials q n and r n is similar to recovery described in (4.33)and (4.34), which are used as the standard recovery formulas for other standardMarchenko systems.Inspired by (3.43) and (3.46) we define the scalar quantities K ( u,v ) nm and¯ K ( p,s ) nm , respectively, as K ( u,v ) nm := ∞ X l = m h K ( u,v ) nl i ∞ X l = n h ¯ K ( u,v ) nl i , m ≥ n, (7.1)¯ K ( p,s ) nm := ∞ X l = m h ¯ K ( p,s ) nl i ∞ X l = n h K ( p,s ) nl i , m ≥ n, (7.2)where we use the notation of (3.42) and recall that K ( u,v ) nl and ¯ K ( u,v ) nl satisfy (f)and (g) of Theorem 2.1, and similarly, K ( p,s ) nl and ¯ K ( p,s ) nl satisfy (a) and (b) ofCorollary 2.2. We remark that the m -dependence of K ( u,v ) nm and ¯ K ( p,s ) nm occursonly in the numerators in (7.1) and (7.2). When m = n, with the help of (3.40),553.41), (7.1), and (7.2) we obtain K ( u,v ) nn = ∞ X l = n h K ( u,v ) nl i ∞ X l = n h ¯ K ( u,v ) nl i = h ψ ( u,v ) n (1) i h ¯ ψ ( u,v ) n (1) i , (7.3)¯ K ( p,s ) nn = ∞ X l = n h ¯ K ( p,s ) nl i ∞ X l = n h K ( p,s ) nl i = h ¯ ψ ( p,s ) n (1) i h ψ ( p,s ) n (1) i . (7.4)Comparing (3.43), (3.46), (7.3), and (7.4) we observe that the potentials q n and r n are recovered from K ( u,v ) nm and ¯ K ( p,s ) nm , respectively, as q n = D ( q,r ) ∞ E ( q,r ) ∞ (cid:16) K ( u,v ) nn − K ( u,v )( n +1)( n +1) (cid:17) , (7.5) r n = E ( q,r ) ∞ D ( q,r ) ∞ (cid:16) ¯ K ( p,s )( n − n − − ¯ K ( p,s ) nn (cid:17) , (7.6)where we recall that D ( q,r ) ∞ and E ( q,r ) ∞ are the constants appearing in (2.50) and(2.51), respectively.In the next theorem we show that the scalar quantities K ( u,v ) nm and ¯ K ( p,s ) nm given in (7.1) and (7.2) satisfy the respective alternate Marchenko equations,for m > n, given by K ( u,v ) nm + ¯ G ( u,v ) n + m + ∞ X l = n +1 ∞ X j = n +1 (cid:16) K ( u,v ) n ( j +1) − K ( u,v ) nj (cid:17) G ( u,v ) j + l (cid:16) ¯ G ( u,v ) l + m − ¯ G ( u,v ) l + m − (cid:17) = 0 , (7.7)¯ K ( p,s ) nm + G ( p,s ) n + m + ∞ X l = n +1 ∞ X j = n +1 (cid:16) ¯ K ( p,s ) n ( j +1) − ¯ K ( p,s ) nj (cid:17) ¯ G ( p,s ) j + l (cid:16) G ( p,s ) l + m − G ( p,s ) l + m − (cid:17) = 0 , (7.8)where we have defined G ( u,v ) n := ∞ X j = n Ω ( u,v ) j , ¯ G ( u,v ) n := ∞ X j = n ¯Ω ( u,v ) j , (7.9) G ( p,s ) n := ∞ X j = n Ω ( p,s ) j , ¯ G ( p,s ) n := ∞ X j = n ¯Ω ( p,s ) j , (7.10)with the scalar functions Ω ( u,v ) j , ¯Ω ( u,v ) j , Ω ( p,s ) j , ¯Ω ( p,s ) j defined as in (4.26) for thepotentials pairs ( u, v ) and ( p, s ) , respectively.56 heorem 7.1. Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to ( q, r ) as in (3.1) – (3.4) . Let K ( u,v ) nm and ¯ K ( p,s ) nm bethe scalar quantities defined as in (7.1) and (7.2) , respectively, and let G ( u,v ) n , ¯ G ( u,v ) n , G ( p,s ) n , ¯ G ( p,s ) n be the quantities defined in (7.9) and (7.10) . Then, K ( u,v ) nm and ¯ K ( p,s ) nm satisfy the alternate Marchenko system given in (7.7) and (7.8) ,respectively.Proof. In the notation of (3.42), the (1 ,
2) entry in the matrix Marchenko system(4.24) for the potential pair ( u, v ) is given by h K ( u,v ) nk i + ¯Ω ( u,v ) n + k + ∞ X l = n +1 h ¯ K ( u,v ) nl i ¯Ω ( u,v ) l + k = 0 , k > n. (7.11)Adding and subtracting ¯Ω ( u,v ) n + k to ¯Ω ( u,v ) l + k in (7.11), we obtain h K ( u,v ) nk i + ¯Ω ( u,v ) n + k + ∞ X l = n +1 h ¯ K ( u,v ) nl i ¯Ω ( u,v ) n + k + ∞ X l = n +1 h ¯ K ( u,v ) nl i (cid:16) ¯Ω ( u,v ) l + k − ¯Ω ( u,v ) n + k (cid:17) = 0 . (7.12)Using [ ¯ K ( u,v ) nn ] = 1 , as seen from the first equality in (2.15), we combine thesecond and third terms on the left-hand side of (7.12) to obtain h K ( u,v ) nk i + ¯Ω ( u,v ) n + k ∞ X l = n h ¯ K ( u,v ) nl i + ∞ X l = n +1 h ¯ K ( u,v ) nl i (cid:16) ¯Ω ( u,v ) l + k − ¯Ω ( u,v ) n + k (cid:17) = 0 . (7.13)From (3.40) we see that the summation in the second term on the left-hand sideof (7.13) is equal to [ ¯ ψ ( u,v ) n (1)] , and hence by dividing (7.13) by that term weget h K ( u,v ) nk i h ¯ ψ ( u,v ) n (1) i + ¯Ω ( u,v ) n + k + ∞ X l = n +1 h ¯ K ( u,v ) nl i h ¯ ψ ( u,v ) n (1) i (cid:16) ¯Ω ( u,v ) l + k − ¯Ω ( u,v ) n + k (cid:17) = 0 . (7.14)Using the (1 ,
1) entry in the Marchenko system (4.24) for the potential pair( u, v ) , we write (7.14) as h K ( u,v ) nk i h ¯ ψ ( u,v ) n (1) i + ¯Ω ( u,v ) n + k − ∞ X l = n +1 ∞ X j = n +1 h K ( u,v ) nj i h ¯ ψ ( u,v ) n (1) i Ω ( u,v ) j + l (cid:16) ¯Ω ( u,v ) l + k − ¯Ω ( u,v ) n + k (cid:17) = 0 . (7.15)57aking the summation for k ≥ m in (7.15) and using (7.9) we get ∞ X k = m h K ( u,v ) nk i h ¯ ψ ( u,v ) n (1) i + ¯ G ( u,v ) n + m − ∞ X l = n +1 ∞ X j = n +1 h K ( u,v ) nj i h ¯ ψ ( u,v ) n (1) i Ω ( u,v ) j + l (cid:16) ¯ G ( u,v ) l + m − ¯ G ( u,v ) n + m (cid:17) = 0 . (7.16)Further, using (3.40), (7.1), and (7.9) in (7.16), for m > n we obtain K ( u,v ) nm + ¯ G ( u,v ) n + m − ∞ X l = n +1 ∞ X j = n +1 (cid:16) K ( u,v ) nj − K ( u,v ) n ( j +1) (cid:17) (cid:16) G ( u,v ) l + j − G ( u,v ) l + j +1 (cid:17) (cid:16) ¯ G ( u,v ) l + m − ¯ G ( u,v ) n + m (cid:17) = 0 . (7.17)It is lengthy but straightforward to show that ∞ X l = n +1 (cid:16) G ( u,v ) l + j − G ( u,v ) l + j +1 (cid:17) (cid:16) ¯ G ( u,v ) l + m − ¯ G ( u,v ) n + m (cid:17) = ∞ X l = n +1 G ( u,v ) j + l (cid:16) ¯ G ( u,v ) l + m − ¯ G ( u,v ) l + m − (cid:17) . (7.18)Finally, using (7.18) in (7.17) we obtain (7.7). The derivation of (7.8) is similarlyobtained with the help of the (2 ,
1) and (2 ,
2) entries of (4.24) for the potentialpair ( p, s ) . In this section we describe various methods to recover the potentials q n and r n when the scattering data set for (1.1) is available. We recall that the scatteringdata set consists of the scattering coefficients and the bound-state information.Because of Theorem 2.6, the four scattering coefficients T ( q,r ) , ¯ T ( q,r ) , R ( q,r ) , ¯ R ( q,r ) contain all the information about the scattering coefficients for (1.1).Similarly, because of Theorem 4.1, (4.18)–(4.23), and (4.41), the matrix triplets( A, B, C ( q,r ) ) and ( ¯ A, ¯ B, ¯ C ( q,r ) ) contain all the information related to the boundstates of (1.1). We let D ( q,r ) := { T ( q,r ) , ¯ T ( q,r ) , R ( q,r ) , ¯ R ( q,r ) , ( A, B, C ( q,r ) ) , ( ¯ A, ¯ B, ¯ C ( q,r ) ) } , (8.1)and refer to D ( q,r ) as the scattering data set for (1.1). Let us mention thatthe relevant constants D ( q,r ) ∞ and E ( q,r ) ∞ are obtained from T ( q,r ) and ¯ T ( q,r ) via(2.59), and hence D ( q,r ) ∞ and E ( q,r ) ∞ are known if D ( q,r ) is known.Using the theory developed in Sections 2–7, we are able to solve the inverseproblem for (1.1) in various ways, and we outline below some of those methods.58a) The standard Marchenko method.
In this method, using the scatter-ing data set D ( q,r ) described in (8.1), we construct the scalar quantitiesΩ ( q,r ) k and ¯Ω ( q,r ) k defined in (6.2) and use them as input to the Marchenkosystem (6.1). It can be proved in the standard way that (6.1) is uniquelysolvable via iteration. From the solution [ ¯ M ( q,r ) nm M ( q,r ) nm ] to (6.1) we re-cover q n and r n via (6.13) and (6.14), respectively.(b) The alternate Marchenko method.
In this method, using the scat-tering data set D ( q,r ) , we first obtain the constants D ( q,r ) ∞ and E ( q,r ) ∞ via(2.59) and also obtain Ω ( q,r ) k and ¯Ω ( q,r ) k defined in (6.2). Then, we con-struct the scalar quantities Ω ( p,s ) k and ¯Ω ( p,s ) k via (6.6). Moreover, using(4.59), (4.60), and (6.6) we construct Ω ( u,v ) k and ¯Ω ( u,v ) k asΩ ( u,v ) k = D ( q,r ) ∞ E ( q,r ) ∞ (cid:16) Ω ( q,r ) k − Ω ( q,r ) k − (cid:17) , ¯Ω ( u,v ) k = E ( q,r ) ∞ D ( q,r ) ∞ ∞ X l =0 ¯Ω ( q,r ) k +2 l . (8.2)Next, we use (7.9) to obtain G ( u,v ) k and ¯ G ( u,v ) k and use (7.10) to get G ( p,s ) k and ¯ G ( p,s ) k . Using G ( u,v ) k and ¯ G ( u,v ) k as input to the uncoupled alternateMarchenko equation (7.7), we obtain K ( u,v ) nm . Similarly, using G ( p,s ) k and¯ G ( p,s ) k as input to the uncoupled alternate Marchenko equation (7.8), weobtain ¯ K ( p,s ) nm . Finally, we recover the potentials q n and r n via (7.5) and(7.6), respectively.(c) Inversion with the help of the Marchenko system for (1.5) . In thismethod, from the scattering data set D ( q,r ) we first obtain the constants D ( q,r ) ∞ and E ( q,r ) ∞ via (2.59) and also obtain Ω ( q,r ) k and ¯Ω ( q,r ) k defined in(6.2). Then, we get Ω ( u,v ) k and ¯Ω ( u,v ) k via (8.2). Using Ω ( u,v ) k and ¯Ω ( u,v ) k asinput to the Marchenko system (4.24), we obtain K ( u,v ) nm and ¯ K ( u,v ) nm . Next,using (3.40) we recover the 2 × ψ ( u,v ) n (1) ψ ( u,v ) n (1)] from K ( u,v ) nm and ¯ K ( u,v ) nm . Finally, we use (3.43) and (3.44) to recover the potentials q n and r n , respectively.(d) Inversion with the help of the Marchenko system for (1.6) . Inthis method, using (2.59) we first obtain the constants D ( q,r ) ∞ and E ( q,r ) ∞ and also obtain Ω ( q,r ) k and ¯Ω ( q,r ) k defined in (6.2) from the scattering dataset D ( q,r ) . Then, we get Ω ( p,s ) k and ¯Ω ( p,s ) k via (6.6). Next, using Ω ( p,s ) k and¯Ω ( p,s ) k as input to the Marchenko system (4.24), we obtain [ ¯ K ( p,s ) nm K ( p,s ) nm ] . Then, via (3.41) we get [ ¯ ψ ( p,s ) n (1) ψ ( p,s ) n (1)] . Finally, we use (3.45) and(3.46) to recover the potentials q n and r n , respectively.(e) Inversion by first recovering the potentials u n and s n . In thismethod, from the scattering data set D ( q,r ) we first obtain the constants D ( q,r ) ∞ and E ( q,r ) ∞ via (2.59) and also obtain Ω ( q,r ) k and ¯Ω ( q,r ) k defined in596.2). Then, we construct Ω ( u,v ) k and ¯Ω ( u,v ) k via (8.2) and also constructΩ ( p,s ) k and ¯Ω ( p,s ) k via (6.6). Next, using Ω ( u,v ) k and ¯Ω ( u,v ) k as input in theuncoupled Marchenko equation given in the first line of (4.31) related to( u, v ), we obtain [ K ( u,v ) nm ] , from which we recover u n as in the first equalityin (4.33). Similarly, using Ω ( p,s ) k and ¯Ω ( p,s ) k as input in the uncoupledMarchenko equation given in the second line of (4.31) related to ( p, s ),we obtain [ ¯ K ( p,s ) nm ] , from which we recover s n as in the second equality in(4.34). Finally, we use (3.11) and (3.12) with input ( u n , s n ) and recoverthe potentials q n and r n . In this section we consider an application of our results in integrable semi-discrete systems, and we provide the solution to the nonlinear system (1.4) viathe method of the inverse scattering transform. This is done by describing thetime evolution of the scattering data for (1.1) and determining the correspondingtime-evolved potentials q n and r n . Hence, each of the methods to solve theinverse problem for (1.1) presented in Section 8 can be used to solve (1.4) if wereplace the scattering data set D ( q,r ) appearing in (8.1) with its time-evolvedversion. In this section, we also present certain solution formulas for (1.4)expressed explicitly in terms of the matrix triplets ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) forthe linear system (1.1). Such solution formulas correspond to the reflectionlessscattering data for (1.1), in which case the corresponding Marchenko integralsystem for (1.1) has separable kernels and hence is solved in closed form byusing standard linear algebraic methods.Let us mention [13] that the system (1.4) is the semi-discrete analog of thenonlinear system iq t + q xx − i ( q r q ) x = 0 ,ir t − r xx − i ( r q r ) x = 0 , (9.1)where q ( x, t ) and r ( x, t ) are the continuous analogs of q n and r n when the latterquantities depend on both n and t . The nonlinear system (9.1) is known as thederivative NLS system or the Kaup-Newell system.It is already known [1, 13] that (1.4) can be derived by imposing the com-patibility condition ˙ X n + X n T n +1 − T n X n = 0 , (9.2)where ( X n , T n ) is the AKNS pair with X n being the 2 × T n is the 2 × T n = − i ( z − z + 1) q n − r n ] z (1 + q n − r n ) i ( z − q n − q n − r n − i ( z − q n z (1 − q n r n ) − ir n − − q n − r n − + i z r n q n − r n i ( z − q n − r n , q n , r n ) . Werecall that an overdot denotes the derivative with respect to t. Let us remarkthat the AKNS pair for a given nonlinear system is not unique. One can usethe transformation Ψ n ˜Ψ n := G n Ψ n , X n ˜ X n := G n X n G − n +1 , T n ˜ T n := ˙ G n G − n + G n T n G − n , for any appropriate invertible matrix G n , and the corresponding compatibilitycondition ˙˜ X n + ˜ X n ˜ T n +1 − ˜ T n ˜ X n = 0 , yields the same integrable nonlinear system that (9.2) yields. Since the choiceof X n is not unique, instead of analyzing the linear systemΨ n = X n Ψ n +1 , one can alternatively analyze the system˜Ψ n = ˜ X n ˜Ψ n +1 . The linear system (1.5) is associated with the integrable nonlinear system i ˙ u n + u n − − u n + u n +1 − u n − u n v n − u n u n +1 v n = 0 ,i ˙ v n − v n − + 2 v n − v n +1 + u n v n − v n + u n v n v n +1 = 0 . (9.3)The AKNS pair ( X n , T n ) for (9.3) consists of the matrix X n appearing as thecoefficient matrix in (1.5) and the matrix T n given by T n = i ( z − − i u n − v n i z u n − − i u n − iz v n − + i v n i (cid:18) − z (cid:19) + iu n v n − . Similarly, the linear system (1.6) is associated with the integrable system (9.3)with ( u n , v n ) replaced by ( p n , s n ) there.In the following theorem we summarize the time evolution of the scatteringdata for (1.5). A proof is omitted because the time evolution of the scatteringcoefficients is described in [13] and the time evolution of the norming constantsfor simple bound states described in [13] is readily generalized to the case ofnon-simple bound states and hence to the time evolution of the matrix triplets. Theorem 9.1.
Assume that the potentials u n and v n appearing in (1.5) and (9.3) are rapidly decaying and − u n v n = 0 for n ∈ Z . Then, the correspondingreflection coefficients evolve in time as R ( u,v ) R ( u,v ) e − it ( z − z − ) , ¯ R ( u,v ) ¯ R ( u,v ) e it ( z − z − ) ,L ( u,v ) L ( u,v ) e it ( z − z − ) , ¯ L ( u,v ) ¯ L ( u,v ) e − it ( z − z − ) , nd the transmission coefficients T ( u,v )l , T ( u,v )r , ¯ T ( u,v )l , ¯ T ( u,v )r do not changein time. Furthermore, in the corresponding matrix triplets ( A, B, C ( u,v ) ) and ( ¯ A, ¯ B, ¯ C ( u,v ) ) describing the bound-state data for (1.5) , the row vectors C ( u,v ) and ¯ C ( u,v ) evolve in time as C ( u,v ) C ( u,v ) e − it ( A − A − ) , ¯ C ( u,v ) ¯ C ( u,v ) e it [ ¯ A − ( ¯ A ) − ] , (9.4) and the matrices A, ¯ A, B, ¯ B do not change in time. Moreover, the constant D ( u,v ) ∞ appearing in (2.10) does not change in time, either. We remark that Theorem 9.1 holds in the same way for the system (1.6) withthe potential pair ( p, s ) . Next, we present the time evolution of the scatteringdata for (1.1).
Theorem 9.2.
Assume that the potentials q n and r n appearing in (1.1) and (1.4) are rapidly decaying and satisfy (1.2) . Then, the corresponding reflectioncoefficients evolve in time as R ( q,r ) R ( q,r ) e − it ( z − z − ) , ¯ R ( q,r ) ¯ R ( q,r ) e it ( z − z − ) ,L ( q,r ) L ( q,r ) e it ( z − z − ) , ¯ L ( q,r ) ¯ L ( q,r ) e − it ( z − z − ) , (9.5) and the corresponding transmission coefficients T ( q,r ) and ¯ T ( q,r ) do not changein time. Furthermore, in the matrix triplets ( A, B, C ( q,r ) ) and ( ¯ A, ¯ B, ¯ C ( q,r ) ) describing the bound-state data for (1.1) , the row vectors C ( q,r ) and ¯ C ( q,r ) evolvein time according to C ( q,r ) C ( q,r ) e − it ( A − A − ) , ¯ C ( q,r ) ¯ C ( q,r ) e it [ ¯ A − ( ¯ A ) − ] . (9.6) Moreover, neither of constants D ( q,r ) ∞ and E ( q,r ) ∞ appearing in (2.50) and (2.51) ,respectively, changes in time.Proof. Let us first prove that D ( q,r ) ∞ does not change in time, i.e. we have˙ D ( u,v ) ∞ = 0 , where we recall that we use an overdot to denote the time derivative.From the second equality in (2.50) and the fact that D ( q,r ) ∞ = 0 , we see that˙ D ( q,r ) ∞ = 0 if and only if ˙ D ( q,r ) ∞ /D ( q,r ) ∞ = 0 , which is equivalent to having ∞ X n = −∞ ˙ q n r n + q n ˙ r n − q n r n = 0 . (9.7)In order to prove that (9.7) holds, we multiply the first line of (1.4) with r n andthe second line of (1.4) with q n and then we add the resulting equations. Usingthe summation over n, after some straightforward simplifications, we get ∞ X n = −∞ ˙ q n r n + q n ˙ r n − q n r n = ∞ X n = −∞ (∆ n +1 − ∆ n ) , (9.8)62here we have let∆ n := i (cid:20)
11 + q n − r n − q n r n − (1 − q n − r n − )(1 − q n r n ) (cid:21) . (9.9)Since the potentials q n and r n are rapidly decaying as n → ±∞ and satisfy (1.2),from (9.9) we see that ∆ n is well defined and rapidly decaying as n → ±∞ . Hence, the telescoping series on the right-hand side of (9.8) converges to 0 , which completes the proof that ˙ D ( q,r ) ∞ = 0 . The proof of ˙ E ( q,r ) ∞ = 0 is obtainedin a similar manner by establishing that ˙ E ( q,r ) ∞ /E ( q,r ) ∞ = 0 , which is equivalentto having ∞ X n = −∞ ˙ q n r n +1 + q n ˙ r n +1 q n r n +1 = 0 . (9.10)In order to prove (9.10), we replace n by n + 1 in the second line of (1.4) andmultiply the resulting equation by q n , and then we add to that equation thefirst line of (1.4) multiplied by r n +1 . Then, a summation over n , after somestraightforward simplifications, yields ∞ X n = −∞ ˙ q n r n +1 + q n ˙ r n +1 q n r n +1 = ∞ X n = −∞ (Θ n +1 − Θ n ) , (9.11)where we have letΘ n := i (cid:20) − q n r n − − q n − r n +1 (1 + q n − r n )(1 + q n r n +1 ) (cid:21) . From the properties of q n and r n , it follows that Θ n is well defined and rapidlydecaying as n → ±∞ . Thus, the telescoping series in (9.11) is convergent to0 , which establishes the proof that ˙ E ( q,r ) ∞ = 0 . When the potential pairs ( q, r ) , ( u, v ) , ( p, s ) are related to each other as in (3.1)–(3.4), we have the matrices A, ¯ A, B, ¯ B appearing in (4.26) are common and the scattering coefficients for(1.1), (1.5), (1.6) are related as described in Theorem 3.4. Thus, with the helpof Theorem 3.4, Theorem 9.1, and the fact that ˙ D ( q,r ) ∞ = 0 and ˙ E ( q,r ) ∞ = 0 , weconclude that the transmission coefficients T ( q,r ) and ¯ T ( q,r ) do not change intime and the reflection coefficients evolve as in (9.5). Furthermore, from (4.46),(9.4), and the fact that ˙ D ( q,r ) ∞ = 0 and ˙ E ( q,r ) ∞ = 0 , we obtain (9.6).Next we consider explicit solutions to the integrable systems (1.4) and (9.3)by using the method of [4, 6]. Such explicit solutions correspond to the zeroreflection coefficients and the time-evolved scattering data sets. From Theo-rem 9.1 and Theorem 9.2 we see that the matrix triplets corresponding to (1.4)and (9.3) have similar time evolutions described as( A, B, C ) ( A, B, C E ) , ( ¯ A, ¯ B, ¯ C ) ( ¯ A, ¯ B, ¯ C ¯ E ) , (9.12)where we have defined E := e − it ( A − A − ) , ¯ E := e it [ ¯ A − ( ¯ A ) − ] . (9.13)63et us remark that (4.24) and (4.26) for the potential pair ( u, v ) and (6.1) and(6.2) for the potential pair ( q, r ) are similar, and hence the solution to (6.1) isobtained in a similar way the solution to (4.24) is obtained.Our goal now is to present the corresponding explicit solutions to (4.24) and(6.1) when their Marchenko kernels are given byΩ n + m = C E A n + m − B, ¯Ω n + m = ¯ C ¯ E ( ¯ A ) − n − m − ¯ B. (9.14)Note that we impose no restriction on the values of N, ¯ N, z j , ¯ z j , m j , ¯ m j in the matrix triplets ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) appearing in (4.18) and (4.21),respectively. Hence, this method yields an enormous number of explicit solutionsto each of (4.24) and (6.1). From (9.14) we see that the Marchenko kernelsΩ n + m and ¯Ω n + m are separable in n and m, i.e. we can write them as thematrix products given byΩ n + m = [ C A n ] (cid:2) E A m − B (cid:3) , ¯Ω n + m = (cid:2) ¯ C ( ¯ A ) − n (cid:3) (cid:2) ¯ E ( ¯ A ) − m − ¯ B (cid:3) , (9.15)where we have used the fact that the matrices A and E commute and that thematrices ¯ A and ¯ E commute.Before we present our explicit solutions to (4.24) and (6.1), we introducesome auxiliary quantities. In terms of the positive integers m j , N, ¯ m j , ¯ N appearing in (4.18)–(4.23), we introduce the positive integers N and ¯ N as N := N X j =1 m j , ¯ N := ¯ N X j =1 ¯ m j . (9.16)From the results in Section 4 it follows that 2( N + ¯ N ) corresponds to thetotal number of bound states including the multiplicities for (1.1) and (1.5). Interms of the matrix triplets ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) let us introduce the N × ¯ N matrix Υ and the ¯ N × N matrix ¯Υ asΥ := ∞ X k =0 A k B ¯ C ( ¯ A ) − k , ¯Υ := ∞ X k =0 ( ¯ A ) − k ¯ B C A k . (9.17)In terms of the two matrix triplets let us also define the N × N matrix U n andthe ¯ N × ¯ N matrix ¯ U n as U n := I − ¯ E ( ¯ A ) − n − ¯Υ E A n +1 Υ ( ¯ A ) − n − , (9.18)¯ U n := I − E A n Υ ¯ E ( ¯ A ) − n − ¯Υ A n +1 , (9.19)where we recall that the N × N matrix E and the ¯ N × ¯ N matrix ¯ E are definedin (9.13).In the next proposition we elaborate on the matrices Υ and ¯Υ . Proposition 9.3.
Let ( A, B, C ) and ( ¯ A, ¯ B, ¯ C ) be the matrix triplets appearingin (4.18) – (4.23) with | z j | < for ≤ j ≤ N and | ¯ z j | > for ≤ j ≤ ¯ N .
Then,the matrices Υ and ¯Υ defined in (9.17) are the unique solutions to the respectivelinear systems Υ − A Υ( ¯ A ) − = B ¯ C, ¯Υ − ( ¯ A ) − ¯Υ A = ¯ B C. (9.20)64 roof.
By premultiplying the first equality in (9.17) by A and postmultiplyingit by ( ¯ A ) − and subtracting the resulting matrix equality from the originalequality, we obtain the first linear system in (9.20). The second equality in(9.20) is similarly obtained from the second equality in (9.17). The existenceand uniqueness of the solutions to the two matrix systems in (9.20) can beanalyzed as in Theorem 18 . A, B, C ) and( ¯ A, ¯ B, ¯ C ) , we have the unique solutions Υ and ¯Υ to (9.20) if and only if theproduct of an eigenvalue of A and an eigenvalue of ( ¯ A ) − is never equal to1 . The satisfaction of the latter condition directly follows from the fact that | z j | < ≤ j ≤ N and | ¯ z j | > ≤ j ≤ ¯ N .
Thus, the solutions Υ and¯Υ to (9.20) are unique and given by (9.17).Next, we present the explicit solution formula for the Marchenko system(4.24) corresponding to the Marchenko kernels given in (9.15) for the potentialpair ( u, v ) . Theorem 9.4.
Using the time-evolved reflectionless Marchenko kernels Ω ( u,v ) n + m and ¯Ω ( u,v ) n + m that have the form as in (9.15) for the potential pair ( u, v ) , the cor-responding Marchenko system (4.24) , in the notation of (3.42) , has the solutiongiven by h K ( u,v ) nm i = − ¯ C ( u,v ) ( ¯ A ) − n ( U ( u,v ) n ) − ¯ E ( ¯ A ) − m − ¯ B, (9.21) h K ( u,v ) nm i = C ( u,v ) A n ( ¯ U ( u,v ) n ) − E A n Υ ( u,v ) ¯ E ( ¯ A ) − m − n − ¯ B, (9.22) h ¯ K ( u,v ) nm i = ¯ C ( u,v ) ( ¯ A ) − n ( U ( u,v ) n ) − ¯ E ( ¯ A ) − n − ¯Υ ( u,v ) E ( A ) n + m B, (9.23) h ¯ K ( u,v ) nm i = − C ( u,v ) A n ( ¯ U ( u,v ) n ) − E A m − B, (9.24) where Υ ( u,v ) and ¯Υ ( u,v ) are the matrices appearing in (9.17) for the potentialpair ( u, v ) and the matrices U ( u,v ) n and ¯ U ( u,v ) n are defined as in (9.18) and (9.19) for the potential pair ( u, v ) . Proof.
For simplicity, we suppress the superscript ( u, v ) in the proof. We alreadyknow that the Marchenko system (4.24) is equivalent to the combination of theuncoupled system (4.31) and the system (4.32). To obtain (9.21) we proceed asfollows. Using (9.15) as input to the first line of (4.31) we get (cid:2) K nm (cid:3) + ¯ C ( ¯ A ) − n ¯ E ( ¯ A ) − m − ¯ B − ∞ X l = n +1 ∞ X j = n +1 (cid:2) K nj (cid:3) C E A j + l − B ¯ C ( ¯ A ) − l ¯ E ( ¯ A ) − m − ¯ B = 0 , (9.25)where we have used the fact that E and A commute and ¯ E and ¯ A commute.From (9.25) we see that [ K nm ] has the form (cid:2) K nm (cid:3) = H n ¯ E ( ¯ A ) − m − ¯ B, (9.26)65here H n satisfies H n I − ∞ X l = n +1 ∞ X j = n +1 ¯ E ( ¯ A ) − j − ¯ B C E A j + l − B ¯ C ( ¯ A ) − l = − ¯ C ( ¯ A ) − n . (9.27)Using (9.17) on the left-hand side of (9.27), we write (9.27) as H n U n = − ¯ C ( ¯ A ) − n , (9.28)where U n is the matrix defined in (9.18). From (9.28) we get H n = − ¯ C ( ¯ A ) − n ( U n ) − , (9.29)and using (9.29) in (9.26) we obtain (9.21). The solution formula for (cid:2) ¯ K nm (cid:3) appearing in (9.24) is obtained in a similar manner by using the second line of(4.31). Then, using (9.15) and (9.24) in the second line of (4.32) we obtain theformula for [ K nm ] given in (9.22). Similarly, by using (9.15) and (9.21) in thefirst line of (4.32), we obtain the formula for [ ¯ K nm ] given in (9.23).We remark that the result of Theorem 9.4 remains valid for the Marchenkosystem (6.1) because of the resemblance between (4.24) and (6.1) and the factthat (9.12) and (9.13) have the same appearance for the potential pairs ( u, v )and ( q, r ) . So, without a proof we state the result in the next corollary.
Corollary 9.5.
Using the time-evolved reflectionless Marchenko kernels Ω ( q,r ) n + m and ¯Ω ( q,r ) n + m that have the form as in (9.15) for the potential pair ( q, r ) , the cor-responding Marchenko system (6.1) , in the notation of (3.42) , has the solutiongiven by h M ( q,r ) nm i = − ¯ C ( q,r ) ( ¯ A ) − n ( U ( q,r ) n ) − ¯ E ( ¯ A ) − m − ¯ B, (9.30) h M ( q,r ) nm i = C ( q,r ) A n ( ¯ U ( q,r ) n ) − E A n Υ ( q,r ) ¯ E ( ¯ A ) − n − m − ¯ B, (9.31) h ¯ M ( q,r ) nm i = ¯ C ( q,r ) ( ¯ A ) − n ( U ( q,r ) n ) − ¯ E ( ¯ A ) − n − ¯Υ ( q,r ) E ( A ) n + m B, (9.32) h ¯ M ( q,r ) nm i = − C ( q,r ) A n ( ¯ U ( q,r ) n ) − E A m − B, (9.33) where Υ ( q,r ) and ¯Υ ( q,r ) are the matrices appearing in (9.17) for the potentialpair ( q, r ) and the matrices U ( q,r ) n and ¯ U ( q,r ) n are defined as (9.18) and (9.19) for the potential pair ( q, r ) . In the next proposition, when (3.1)–(3.4) hold, we show how some relevantquantities for the potential pairs ( u, v ) and ( p, s ) are related to the correspondingquantities for the potential pair ( q, r ) . These results will enable us to obtainexplicit solutions to the nonlinear system (1.4) by using the input data directlyrelated to the potential pair ( q, r ) . roposition 9.6. Assume that the potentials q n and r n appearing in (1.1) arerapidly decaying and satisfy (1.2) . Assume further that the potential pairs ( u, v ) and ( p, s ) are related to the potential pair ( q, r ) as in (3.1) – (3.4) . Then, we havethe following: (a) The matrices Υ and ¯Υ appearing in (9.17) corresponding to the potentialpairs ( u, v ) and ( p, s ) are related to those with the potential pair ( q, r ) as Υ ( u,v ) = E ( q,r ) ∞ D ( q,r ) ∞ Υ ( q,r ) (cid:2) I − ( ¯ A ) − (cid:3) − , ¯Υ ( u,v ) = D ( q,r ) ∞ E ( q,r ) ∞ ¯Υ ( q,r ) (cid:0) I − A − (cid:1) , (9.34)Υ ( p,s ) = E ( q,r ) ∞ D ( q,r ) ∞ Υ ( q,r ) , ¯Υ ( p,s ) = D ( q,r ) ∞ E ( q,r ) ∞ ¯Υ ( q,r ) , (9.35) where D ( q,r ) ∞ and E ( q,r ) ∞ are the constants appearing in (2.50) and (2.51) ,respectively. (b) The matrices U n and ¯ U n appearing in (9.18) and (9.19) , respectively, cor-responding to the potential pairs ( u, v ) and ( p, s ) are related to the quan-tities relevant to the potential pair ( q, r ) as U ( u,v ) n = I + ¯ E ( ¯ A ) − n − ¯Υ ( q,r ) E A n − (cid:0) I − A (cid:1) Υ ( q,r ) ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − , (9.36)¯ U ( u,v ) n = I + E A n Υ ( q,r ) ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − ¯Υ ( q,r ) A n − (cid:0) I − A (cid:1) , (9.37) U ( p,s ) n = U ( q,r ) n , ¯ U ( p,s ) n = ¯ U ( q,r ) n , (9.38) where Υ ( q,r ) and ¯Υ ( q,r ) are the matrices appearing in (9.17) for the poten-tial pair ( q, r ) . Proof.
Using (4.46) in (9.17) we get (9.34). Similarly, using (4.45) in (9.17) wehave (9.35). Next, using (9.34) in (9.18) we obtain (9.36). Then, using (9.34)in (9.19) we get (9.37). Finally, using (9.35) in (9.18) and (9.19) we obtain(9.38).In the next theorem we present the explicit solution formulas for the alternateMarchenko equations (7.7) and (7.8) corresponding to the time-evolved reflec-tionless scattering data expressed in terms of the matrix triplets (
A, B, C ( u,v ) ) , ( ¯ A, ¯ B, ¯ C ( u,v ) ) , ( A, B, C ( p,s ) ) , and ( ¯ A, ¯ B, ¯ C ( p,s ) ) . heorem 9.7. Using as input the time-evolved reflectionless Marchenko kernels Ω ( u,v ) n + m and ¯Ω ( u,v ) n + m that have the form as in (9.15) for the potential pair ( u, v ) , the corresponding alternate Marchenko equation (7.7) has the explicit solutiongiven by K ( u,v ) nm = − ¯ C ( u,v ) ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − ( V ( u,v ) n ) − ¯ E ( ¯ A ) − m ¯ B, (9.39) where the ¯ N × ¯ N matrix V ( u,v ) n is defined as V ( u,v ) n := I + ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) ¯Υ ( u,v ) E A n +1 ( I − A ) − Υ ( u,v ) ( ¯ A ) − n − , with ¯ N being the positive integer defined in (9.16) and with Υ ( u,v ) and ¯Υ ( u,v ) being the matrices appearing in (9.17) for the potential pair ( u, v ) . Similarly, us-ing as input the time-evolved reflectionless Marchenko kernels Ω ( p,s ) n + m and ¯Ω ( p,s ) n + m that have the form as in (9.15) for the potential pair ( p, s ) , the correspondingalternate Marchenko equation (7.8) has the explicit solution given by ¯ K ( p,s ) nm = − C ( p,s ) A n − ( I − A ) − ( ¯ V ( p,s ) n ) − E A m B, (9.40) where ¯ V ( p,s ) n is the N × N matrix defined as ¯ V ( p,s ) n := I + E A n +1 ( I − A )Υ ( p,s ) ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − ¯Υ ( p,s ) A n − , with N being the positive integer defined in (9.16) and with Υ ( p,s ) and ¯Υ ( p,s ) being the matrices appearing in (9.17) for the potential pair ( p, s ) . Proof.
Using (9.14) with the potential pair ( u, v ) , from (7.9) we obtain G ( u,v ) n = ∞ X k = n C ( u,v ) E A k − B, which is equivalent to G ( u,v ) n = C ( u,v ) E A n − ( I − A ) − B. (9.41)Let us remark that ( I − A ) − is well defined because | z j | < ≤ j ≤ N, asseen from (4.19) and Theorem 4.1. In the same way, from (7.10) and (9.14) weget ¯ G ( u,v ) n = ∞ X k = n ¯ C ( u,v ) ¯ E ( ¯ A ) − k − ¯ B, or equivalently ¯ G ( u,v ) n = ¯ C ( u,v ) ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − ¯ B. (9.42)Using (9.41) and (9.42) in (7.7) and by proceeding in a similar way as in theproof of Theorem 9.4, we obtain (9.39). The explicit solution given in (9.40)is obtained similarly by using in (7.8) the analogs of (9.41) and (9.42) for thepotential pair ( p, s ) . u n and v n are expressed explicitly in terms of the matrix triplets ( A, B, C ( u,v ) ) and( ¯ A, ¯ B, ¯ C ( u,v ) ) as u n = − ¯ C ( u,v ) ( ¯ A ) − n (cid:16) U ( u,v ) n (cid:17) − ¯ E ( ¯ A ) − n − ¯ B,v n = − C ( u,v ) A n (cid:16) ¯ U ( u,v ) n (cid:17) − E A n +1 B, (9.43)where E and ¯ E are the matrices defined in (9.13), and U ( u,v ) n and ¯ U ( u,v ) n arethe matrices appearing in (9.18) and (9.19), respectively, for the potential pair( u, v ) . Let us finally discuss explicit solutions to the nonlinear system (1.4). Wecan express any time-evolved reflectionless scattering data for the potential pair( q, r ) in terms of the Marchenko kernels Ω ( q,r ) n + m and ¯Ω ( q,r ) n + m appearing in (9.14).Hence, as seen from (9.13) and (9.14), we can explicitly determine the corre-sponding solution to (1.4), where q n and r n are explicitly expressed in termsof the matrix triplets ( A, B, C ( q,r ) ) and ( ¯ A, ¯ B, ¯ C ( q,r ) ) . In fact, using these twomatrix triplets as input in any of the inversion methods outlined in Section 8,we are able to obtain explicit solution formulas for (1.4).For example, using these two matrix triplets on the right-hand sides of(9.30)–(9.33), we first obtain the four scalar quantities [ M ( q,r ) nm ] , [ M ( q,r ) nm ] , [ ¯ M ( q,r ) nm ] , [ ¯ M ( q,r ) nm ] , and use them in (6.13) and (6.14) to obtain the solution( q n , r n ) to (1.4) explicitly displayed in terms of the matrix triplets ( A, B, C ( q,r ) )and ( ¯ A, ¯ B, ¯ C ( q,r ) ) . We can obtain another explicit solution formula for (1.4) by expressing theright-hand sides of (7.5) and (7.6) in terms of the matrix triplets (
A, B, C ( q,r ) )and ( ¯ A, ¯ B, ¯ C ( q,r ) ) . That formula is given by q n = τ n − τ n +1 , r n = ¯ τ n − − ¯ τ n , where we have defined τ n := D ( q,r ) ∞ E ( q,r ) ∞ K ( u,v ) nn , ¯ τ n := E ( q,r ) ∞ D ( q,r ) ∞ ¯ K ( p,s ) nn , (9.44)and the right-hand sides in (9.44) are expressed in terms of the quantities rel-evant to the potential pair ( q, r ) with the help of (4.45), (4.46), (9.34), (9.35),(9.39), (9.40). We get τ n = − ¯ C ( q,r ) (cid:2) I − ( ¯ A ) − (cid:3) − (cid:2) I − ( ¯ A ) − (cid:3) − ( ¯ A ) − n − (cid:16) V ( q,r ) n (cid:17) − ¯ E ( ¯ A ) − n ¯ B, ¯ τ n = − C ( q,r ) A n − ( I − A ) − (cid:16) ¯ V ( q,r ) n (cid:17) − E A n B, V ( q,r ) n := I + ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) ¯Υ ( q,r ) ( I − A − ) E A n +1 × ( I − A ) − Υ ( q,r ) (cid:2) I − ( ¯ A − ) (cid:3) − ( ¯ A ) − n − , ¯ V ( q,r ) n := I + E A n +1 ( I − A )Υ ( q,r ) ¯ E ( ¯ A ) − n − (cid:2) I − ( ¯ A ) − (cid:3) − ¯Υ ( q,r ) A n − , with Υ ( q,r ) and ¯Υ ( q,r ) denoting the matrices in (9.17) for ( q, r ) . We can also obtain an explicit solution formula for (1.4) by using q n and r n given in (3.43) and (3.44), respectively, after expressing their right-hand sidesin terms of the quantities relevant to the potential pair ( q, r ) , and this can beachieved with the help of (3.40), (4.46), (9.21)–(9.24), (9.36), and (9.37). Ina similar way, it is possible to obtain an explicit solution formula by using q n and r n given in (3.45) and (3.46), respectively, after expressing their right-handsides in terms of the quantities relevant to the potential pair ( q, r ) . Still anothersolution formula for (1.4) is obtained via (3.11) and (3.12), and this is done asfollows. We first express the right-hand side of the first line of (9.43) in termsof the matrix triplet for the potential pair ( q, r ) , and hence recover u n in termsof the quantities relevant to ( q, r ) . In a similar way, we use the analog of thesecond line of (9.43) for the potential pair ( p, s ) and obtain s n in terms of thequantities relevant to ( q, r ) . Finally, we use the resulting expressions for u n and s n on the right-hand sides of (3.11) and (3.12) and obtain a solution formulafor q n and r n as a solution to (1.4). References [1] M. J. Ablowitz and P. A. Clarkson,
Solitons, nonlinear evolution equationsand inverse scattering , Cambridge Univ. Press, Cambridge, 1991.[2] M. J. Ablowitz, B. Prinari, and A. D. Trubatch,
Discrete and continuousnonlinear Schr¨odinger systems , Cambridge Univ. Press, Cambridge, 2003.[3] T. Aktosun, T. Busse, F. Demontis, and C. van der Mee,
Symmetries for ex-act solutions to the nonlinear Schr¨odinger equation , J. Phys. A , 025202(2010).[4] T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to thefocusing nonlinear Schr¨odinger equation , Inverse Problems , 2171–2195(2007).[5] T. Aktosun and R. Ercan, Direct and inverse scattering problems for afirst-order system with energy-dependent potentials , Inverse Problems ,085002 (2019).[6] T. Aktosun and C. van der Mee, Explicit solutions to the Korteweg-de Vriesequation on the half line , Inverse Problems , 2165–2174 (2006).707] T. N. Busse, Generalized inverse scattering transform for the nonlinearSchr¨odinger equation , Ph.D. thesis, The University of Texas at Arlington,2008.[8] T. N. Busse Martines,
Generalized inverse scattering transform for the non-linear Schr¨odinger equation for bound states with higher multiplicities , Elec-tron. J. Differential Equations, Vol. 2017 (2017), No. 179, pp. 1–15.[9] H. Dym,
Linear algebra in action,
Am. Math. Soc., Providence, RI, 2006.[10] R. Ercan,
Scattering and inverse scattering on the line for a first-ordersystem with energy-dependent potentials , Ph.D. thesis, The University ofTexas at Arlington, 2018.[11] T. Tsuchida,