aa r X i v : . [ qu a n t - ph ] J un Transfer matrix for long-range potentials
Farhang Loran ∗ and Ali Mostafazadeh †∗ Department of Physics, Isfahan University of Technology,Isfahan 84156-83111, Iran † Departments of Mathematics and Physics, Ko¸c University,34450 Sarıyer, Istanbul, Turkey
Abstract
We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials v ( x ) and derive its basic properties. We outline a dynamical formulationof the time-independent scattering theory for this class of potentials where we identify theirtransfer matrix with the S -matrix of a certain effective non-unitary two-level quantum system.For sufficiently large values of | x | , we express v ( x ) as the sum of a short-range potential andan exactly solvable long-range potential. Using this result and the composition property ofthe transfer matrix, we outline an approximation scheme for solving the scattering problem for v ( x ). To demonstrate the effectiveness of this scheme, we construct an exactly solvable long-range potential and compare the exact values of its reflection and transmission coefficientswith those we obtain using our approximation scheme. Transfer matrices have been introduced and employed as a powerful tool for dealing with scatteringproblems for finite-range locally periodic potentials [1, 2, 3]. These typically arise in the study ofwave propagation in multilayered media [4, 5, 6, 7]. By definition, a function v : R → C is called ashort-range potential [8], if there are positive real numbers C , D , and α such that α > | v ( x ) | ≤ C (1 + | x | ) α for | x | ≥ D. (1)An important consequence of this condition is that, as x → ±∞ , the solutions of the time-independent Schr¨odinger equation, − ψ ′′ ( x ) + v ( x ) ψ ( x ) = k ψ ( x ) , (2)tend to plane waves, i.e., ψ ( x ) → A ± e ikx + B ± e − ikx for x → ±∞ . (3) ∗ E-mail address: [email protected] † E-mail address: [email protected] v as a 2 × M that satisfies (cid:20) A + B + (cid:21) = M (cid:20) A − B − (cid:21) . (4)This condition determines M in a unique manner provided that we demand that it is independentof A − and B − [9].The transfer matrix has two important properties [10, 11]:1. Its entries, M ij , determine the left/right reflection ( R l/r ) and transmission ( T l/r ) amplitudesaccording to R l = − M M , R r = M M , T l = T r = T := 1 M . (5)2. For any strictly increasing finite sequence of real numbers, a , a , a , · · · , a n − , and potentials v j defined by v j ( x ) := ( v ( x ) for x ∈ ( a j − , a j ) , , (6)with j ∈ { , , · · · , n } , a := −∞ , and a n := + ∞ , we can express M in terms of the transfermatrices M j of v j according to M = M n M n − · · · M . (7)Equation (7), which is known as the composition property of the transfer matrix, allows for thereduction of the solution of the scattering problem for a given scatterer to that of its slices alongthe scattering axis. This is the main reason for the practical significance of the transfer matrix[6] and its generalizations to multichannel [12, 13, 14, 15, 16], multidimensional [17, 18, 19], andelectromagnetic [20, 21, 22, 23, 24, 25, 26] scattering problems.A recent observation regarding the possibility of reducing scattering problems defined on thehalf-line to those defined on the whole line [27] extends the domain of application of the transfermatrix (4) to potentials defined on the half-line. The principle example of the latter is the effectivepotentials one encounters in solving the Schr¨odinger and Helmholtz equations for a sphericallysymmetric potential in three dimensions, i.e., v eff ( r ) := l ( l + 1) r + v int ( r ) , (8)where r is the radial spherical coordinate, l is the angular momentum quantum number, and v int is the interaction potential. If v int is a finite-range piecewise constant potential, one can expressthe solution of the corresponding Schr¨odinger equation in the intervals where v int ( r ) is constant interms of spherical Bessel and Hankel functions. By letting these play the role of the plane waves, e ± ikx , in the above discussion of the transfer matrix, one can introduce a transfer matrix capable ofdealing with the scattering problem for (8), [28]. See also [29].The present investigation aims at providing a systematic generalization of the notion of transfermatrix to the class C α> / of long-range potentials that satisfy (1) for some α > /
2. It is moti-vated by the above-mentioned developments related to finite-range piecewise constant spherically2ymmetric potentials as well as the recent discovery of long-range potentials supporting full-bandunidirectional invisibility [30, 31, 32, 33].The organization of the paper is as follows. In Sec. 2 we reexamine the transfer matrix for ashort-range potential, explore its relationship with the classical notion of the fundamental matrix ofthe theory of ordinary differential equations, and introduce its decomposition into a pair of matricesthat respectively carry the information about the scattering properties of the potential for left- andright-incident waves. In Sec. 3, we give the definition of the transfer matrix for the real long-range potentials v belonging to C α> / , derive its basic properties, and examine its generalizationto complex long-range potentials. In Sec. 4, we introduce a decomposition of v into the sum ofa short-range potential and an exactly solvable long-range potential. This forms the basis of anapproximation scheme for the solution of the scattering problem for v which we outline in Sec. 5. InSec. 6, we construct an exactly solvable long-range potential and compare the exact expression forits reflection and transmission coefficients with the outcome of our approximation scheme. Finally,in Sec. 7, we present our concluding remarks. Because the Schr¨odinger equation (2) is a second-order linear homogeneous equation, its generalsolution ψ (for each k ) is a linear combination of a pair of linearly independent solutions, ψ and ψ ; ψ ( x ) = c ψ ( x ) + c ψ ( x ) , (9)where c and c are constant coefficients. Introducing the fundamental matrix [34], F ( x ) := (cid:20) ψ ( x ) ψ ( x ) ψ ′ ( x ) ψ ′ ( x ) (cid:21) , (10)we can use (9) to show that (cid:20) ψ ( x ) ψ ′ ( x ) (cid:21) = F ( x ) (cid:20) c c (cid:21) . This in turn implies that for any pair of real numbers x ± , (cid:20) ψ ( x + ) ψ ′ ( x + ) (cid:21) = F ( x + ) F ( x − ) − (cid:20) ψ ( x − ) ψ ′ ( x − ) (cid:21) . (11)Next, we consider the case where v is a short-range potential and examine the consequences ofmaking x ± approach ±∞ . In this case, ψ ( x ± ) tends to A ± e ikx + B ± e − ikx . Therefore, (cid:20) ψ ( x ± ) ψ ′ ( x ± ) (cid:21) → F ( x ± ) (cid:20) A ± B ± (cid:21) for x ± → ±∞ , (12)where F ( x ) := (cid:20) e ikx e − ikx ike ikx − ike − ikx (cid:21) = F (0) e ikx σ , (13)3nd σ i stands for the i -th Pauli matrix. We can use (11) and (12) to relate (cid:20) A − B − (cid:21) to (cid:20) A + B + (cid:21) .This reproduces (4) with the transfer matrix given by M = G (+ ∞ ) G ( −∞ ) − , (14)where G ( ±∞ ) := lim x →±∞ G ( x ) , (15) G ( x ) := F ( x ) − F ( x ) . (16)Substituting (10) and (13) in (16), we can identify the j -th column of G ( x ) with the two-componentwave function: Ψ j ( x ) := F ( x ) − " ψ j ( x ) ψ ′ j ( x ) = 12 " e − ikx { ψ j ( x ) − ik − ψ ′ j ( x ) } e ikx { ψ j ( x ) + ik − ψ ′ j ( x ) } , (17)i.e., G ( x ) = (cid:2) Ψ ( x ) Ψ ( x ) (cid:3) . (18)An interesting property of G ( x ) is that its determinant is proportional to the Wronskian of thesolutions ψ and ψ ; det G ( x ) = i k [ ψ ( x ) ψ ′ ( x ) − ψ ( x ) ψ ′ ( x )] . (19)By virtue of Abel’s theorem [34], this implies that det G ( x ) does not depend on x . Furthermore,because ψ and ψ are linearly independent solutions of the Schr¨odinger equation, their Wronskianand consequently det G ( x ) do not vanish [34].The analysis leading to the decomposition (14) of the transfer matrix is valid for every linearlyindependent pair of solutions, ψ and ψ , of the Schr¨odinger equation (2). If we identify theserespectively with the left- and right-incident scattering solutions, ψ l and ψ r , which by definitionsatisfy ψ l ( x ) → (cid:26) e ikx + R l e − ikx for x → −∞ ,T l e ikx for x → + ∞ , (20) ψ r ( x ) → (cid:26) T r e − ikx for x → −∞ ,e − ikx + R r e ikx for x → + ∞ , (21)and make use of (15) – (18) to compute G ( ±∞ ), we find G ( −∞ ) = (cid:20) R l T r (cid:21) , G (+ ∞ ) = (cid:20) T l R r (cid:21) . (22)Because G ( x ) has a constant nonzero determinant, these equations imply the transmission reci-procity, T r = T l , (23)and the impossibility of perfect absorption [11], T ± = 0 . (24)4ccording to (22) and (23), G ( −∞ ) and G (+ ∞ ) store the scattering properties of the potential forthe left- and right-incident waves, respectively.Substituting (22) and (23) in (14), we arrive at the well-known formula [10, 11]: M = 1 T (cid:20) T − R l R r R r − R l (cid:21) , (25)where T labels the common value of T l and T r . Equations (5) and the fact that det M = 1 followas simple consequences of (25). This provides an alternative verification of item 1 in the list ofproperties of the transfer matrix that we have given in Sec. 1. Ref. [11] outlines the standardderivation of the composition property (7) of transfer matrices which is the content of item 2 of thislist. In the following we pursue an alternative route for establishing this property which is in linewith the dynamical formulation of the (short-range) potential scattering [42, 43].Consider the two-component wave function,Ψ( x ) := F ( x ) − " ψ ( x ) ψ ′ ( x ) = 12 " e − ikx [ ψ ( x ) − ik − ψ ′ ( x )] e ikx [ ψ ( x ) + ik − ψ ′ ( x )] , (26)where ψ is a general solution of the Schr¨odinger equation (2) for the short-range potential v . Dif-ferentiating both sides of Ψ and making use of (2), we find that Ψ satisfies the time-dependentSchr¨odinger equation, i Ψ ′ ( x ) = H ( x )Ψ( x ) , (27)where x plays the role of time, H ( x ) is the non-stationary matrix Hamiltonian, H ( x ) := v ( x )2 k (cid:20) e − ikx − e ikx − (cid:21) = v ( x )2 k e − ikx σ K e ikx σ , (28)and K := i σ + σ = (cid:20) − − (cid:21) . (29)Furthermore, according to (12) and (26),Ψ( x ) → (cid:20) A ± B ± (cid:21) for x → ±∞ . (30)This relation together with (4) and (27) allow us to identify M with U ( ∞ , −∞ ), where U ( x, x )is the evolution operator associated with the Hamiltonian H ( x ) and the initial ‘time’ x , [42]. Because we can express U ( x, x ) as the time-ordered exponential of H ( x ), i.e., U ( x, x ) = T exp (cid:26) − i Z xx H ( s ) ds (cid:27) (31)= I + ∞ X ℓ =1 ( − i ) ℓ Z xx dx ℓ Z x ℓ x dx ℓ − · · · Z x x dx H ( x ℓ ) H ( x ℓ − ) · · · H ( x ) , By definition, U ( x, x ) satisfies Ψ( x ) = U ( x, x )Ψ( x ) for all x ∈ R . In particular, U ( x , x ) = I .
5e have M = U ( ∞ , −∞ ) = T exp (cid:26) − i Z ∞−∞ H ( x ) dx (cid:27) . (32)The transfer matrix possesses the composition property (7), because evolution operators satisfy thesemi-group multiplication rule, U ( x , x ) = U ( x , x ) U ( x , x ) for all x , x , x ∈ R , and the factthat H ( x ) vanishes for values of x for which v ( x ) = 0, [42, 43]. Another notable consequence of(32) is that because H ( x ) is traceless, U ( x, x ) and consequently M have a unit determinant.If we identify H ( x ) with the interaction Hamiltonian for a two-level quantum system, M whichis equal to U ( ∞ , −∞ ) gives the S -matrix of this system [44]. Note however that H ( x ) is manifestlynon-Hermitian (and non-diagonalizable) even if v is a real-valued potential. In the latter case, itis σ -pseudo-Hermitian [45], i.e., H ( x ) † = σ H ( x ) σ − . If v is a complex potential, H ( x ) is σ -pseudo-normal, i.e., it commutes with its σ -pseudo-adjoint, H ( x ) ♯ := σ − H ( x ) † σ , [45].Because ψ and ψ are solutions of the Schr¨odinger equation (2), the corresponding two-componentwave functions, Ψ and Ψ , solve (27). In light of (18), this implies that i G ′ ( x ) = H ( x ) G ( x ) . (33)Equivalently, we have G ( x ) = U ( x, x ) G ( x ) , (34)which gives rise to U ( x, x ) = G ( x ) G ( x ) − . (35)Letting x → −∞ and x → + ∞ in this relation and using (14), we recover (32). Note also thatEqs. (33) – (35) are valid for arbitrary choices of the linearly-independent solutions ψ and ψ ofthe Schr¨odinger equation (2); they need not coincide with the scattering solutions ψ l and ψ r . Let α be a real number and C α>α denote the class of potentials satisfying (1) for some α > α , sothat C α> specifies the set of short-range potentials. The scattering theory of the latter is a well-established mathematical discipline [35]. Extending this theory to encompass long-range potentialshas been an active area of research since the 1960’s [36, 37, 38, 39, 40, 41]. This has primarily beenmotivated by the indisputable physical importance of long-range interactions, such as the Coulombinteraction. The recent discovery of the application of complex long-range potentials in realizingunidirectional invisibility for all frequencies [31, 33] has also drawn attention to the scattering theoryof complex long-range potentials.For real-valued potentials v belonging to C α> / , the absolutely continuous spectrum of theSchr¨odinger operator − ∂ x + v ( x ) coincides with [0 , ∞ ) and its generalized eigenfunctions have theasymptotic WKB form [40, 41]: ψ ( x ) → ˘ A ± e iS ( x ) + ˘ B ± e − iS ( x ) as x → ±∞ , (36) We can extend the definition of C α and notions of short- and long-range potentials to d dimensions by identifyingthe independent variable x in (1) with an element of R d . A ± and ˘ B ± are constant coefficients, S ( x ) := kx + ς ( x ) = kx (cid:20) V ( x )2 k (cid:21) , (37) ς ( x ) := − k Z x v ( s ) ds, (38)and V ( x ) := − x R x v ( s ) ds . It is not difficult to show that V belongs to C α> / . In particular, S tends to an increasing function of x as x → ±∞ . This in turn allows for identifying e iS ( x ) and e − iS ( x ) respectively with asymptotic right- and left-going waves, and suggests defining the transfermatrix of v as the 2 × M that satisfies, " ˘ A + ˘ B + = ˘ M " ˘ A − ˘ B − , (39)and is independent of ˘ A − and ˘ B − .If we identify the scattering solutions, ˘ ψ l and ˘ ψ r , of the Schr¨odinger equation with those fulfillingthe asymptotic boundary conditions:˘ ψ l ( x ) → ( e iS ( x ) + ˘ R l e − iS ( x ) for x → −∞ , ˘ T l e iS ( x ) for x → + ∞ , (40)˘ ψ r ( x ) → ( ˘ T r e − iS ( x ) for x → −∞ ,e − iS ( x ) + ˘ R r e iS ( x ) for x → + ∞ , (41)and identify the reflection and transmission amplitudes of the potential with the coefficients ˘ R l/r and ˘ T l/r appearing in these relations, we are led to the following analog of (5).˘ R l = − ˘ M ˘ M , ˘ R r = ˘ M ˘ M , ˘ T l/r = ˘ T := 1˘ M . (42)A direct implementation of this prescription to short-range real potentials shows that the transfermatrix ˘ M and the reflection and transmission amplitudes, ˘ R l/r and ˘ T l/r , differ from the standardtransfer matrix M and the reflection and transmission amplitudes, R l/r and T l/r . This is simplybecause for a short-range potential, ϑ ± := ς ( ±∞ ) := lim x →±∞ ς ( x ) (43)are finite but not necessarily zero. As a result, (36) would agree with (3) provided that˘ A ± = e − iϑ ± A ± , ˘ B ± = e iϑ ± B ± . (44)In view of (4), (39), and (44), M and ˘ M are related via M = e iϑ + σ ˘ M e − iϑ − σ . (45)This equation together with (5) and (42) imply R l = e − iϑ − ˘ R l , R r = e iϑ + ˘ R r , T = e i ( ϑ + − ϑ − ) ˘ T . (46)7e can similarly introduce a transfer matrix for complex-valued potentials belonging to C α> / provided that | e ± iS ( x ) | tend to finite values as x → ±∞ . This restricts the imaginary part ofthe potential to be short-range. In what follows we confine our discussion to this class of complexlong-range potentials, i.e., consider complex-valued potentials v such thatRe( v ) ∈ C α> / , Im( v ) ∈ C α> , (47)where ‘Re’ and ‘Im’ stand for the real and imaginary parts of their argument, respectively.Let, ϑ ± i := lim x →±∞ Im[ ς ( x )] . Then in view of (47), ϑ ± i are real numbers, and | e iS ( x ) | → e − ϑ ± i for x → ±∞ . Therefore, the e ± iS ( x ) ,that appear in (40) and (41) are not generally unimodular. This would be in conflict with theidentification of | ˘ R l/r | and | ˘ T l/r | with the reflection and transmission coefficients, because thesecoefficients are respectively defined as the ratio of the intensity of the reflection and transmittedwaves to the intensity of the incident wave [46]. To avoid this conflict, we introduce S ± ( x ) := S ( x ) − iϑ ± i , (48)and express the asymptotic expression for the scattering solutions of the Schr¨odinger equation inthe form ψ l ( x ) → (cid:26) e iS − ( x ) + R l e − iS − ( x ) for x → −∞ , T l e iS + ( x ) for x → + ∞ , (49) ψ r ( x ) → (cid:26) T r e − iS − ( x ) for x → −∞ ,e − iS + ( x ) + R r e iS + ( x ) for x → + ∞ , (50)where R l/r and T l/r are respectively the left/right reflection and transmission amplitudes.Relations (49) and (50) suggest that we express the asymptotic form of the general solution ofthe Schr¨odinger equation (2) for complex potentials subject to the conditions (47) as ψ ( x ) → A ± e iS ± ( x ) + B ± e − iS ± ( x ) as x → ±∞ , (51)where A ± and B ± are constant coefficients. Comparing (36) and (51), we observe that A ± = e − ϑ ± i ˘ A ± , B ± = e ϑ ± i ˘ B ± . (52)We identify the transfer matrix for this class of complex potentials with the 2 × M satisfying (cid:20) A + B + (cid:21) = M (cid:20) A − B − (cid:21) . (53)Again, we can relate the reflection and tranmission amplitudes, R l/r and T l/r , to the entries of M ; R l = − M M , R r = M M , T l/r = T := 1 M . (54)With the help of (39), (52), and (53), we can express M in terms of ˘ M according to M = e − ϑ + i σ ˘ M e ϑ − i σ . (55)8his equation together with (42) and (54) imply R l = − e ϑ − i ˘ M ˘ M = e ϑ − i ˘ R l , R r = e − ϑ + i ˘ M ˘ M = e − ϑ + i ˘ R r , T := e ϑ − i − ϑ + i ˘ M = e ϑ − i − ϑ + i ˘ T . (56)It is also not difficult to show that the standard transfer matrix M for short-range complex potentialsis given by M = e iϑ + σ ˘ M e − iϑ − σ = e iϑ + r σ M e − iϑ − r σ , (57)where ϑ ± r := Re( ϑ ± ).Next, we explore the relationship between the transfer matrix ˘ M and the classical notion of afundamental matrix of a second order ordinary differential equation. To do this, we introduce:˘ F ( x ) := (cid:20) e iS ( x ) e − iS ( x ) ike iS ( x ) − ike − iS ( x ) (cid:21) = F (0) e iS ( x ) σ , (58)˘Ψ( x ) := ˘ F ( x ) − (cid:20) ψ ( x ) ψ ′ ( x ) (cid:21) = 12 (cid:20) e − iS ( x ) { ψ ( x ) − ik − ψ ′ ( x ) } e iS ( x ) { ψ ( x ) + ik − ψ ′ ( x ) } (cid:21) , (59)˘Ψ j ( x ) := ˘ F ( x ) − " ψ j ( x ) ψ ′ j ( x ) , (60)˘ G ( x ) := ˘ F ( x ) − F ( x ) = [ ˘Ψ ( x ) ˘Ψ ( x ) ] , (61)where ψ is the general solution of the Schr¨odinger equation (2), ψ j with j ∈ { , } are linearly-independent solutions of this equation, and F ( x ) is the corresponding fundamental matrix (10).We can use (36) to show that ˘Ψ( x ) → " ˘ A ± ˘ B ± for x → ±∞ . (62)This relation together with (11), (39), (59), (60), and (61) imply˘ M = ˘ G (+ ∞ ) ˘ G ( −∞ ) − . (63)Substituting this in (55), we find M = e − ϑ + i σ ˘ G (+ ∞ ) ˘ G ( −∞ ) − e ϑ − i σ . (64)If we respectively identify ψ and ψ with the scattering solutions (40) and (41), we obtain (22) –(25) with G ( ±∞ ), R l/r , T l/r , and M replaced with ˘ G ( ±∞ ), ˘ R l/r , ˘ T l/r , and ˘ M . Together with (55),this provides an alternative derivation of (54) and shows that the transfer matrices ˘ M and M shareProperty 1 of the transfer matrix of the short-range potentials that we have listed in Sec. 1. Thesame holds for Property 2. As we show in the sequel, this follows from the fact that ˘ M coincideswith the S -matrix of an associated effective two-level quantum system.In order to derive the composition property of ˘ M , we first use (2), (36), (37), and (59) to showthat ˘Ψ satisfies i ˘Ψ ′ ( x ) = ˘ H ( x ) ˘Ψ( x ) , (65)9here ˘ H ( x ) := v ( x )2 k (cid:20) e − iS ( x ) − e iS ( x ) (cid:21) = iv ( x )2 k e − iS ( x ) σ σ e iS ( x ) σ . (66)In view of (39), (62), and (65),˘ M = ˘ U ( ∞ , −∞ ) = T exp (cid:26) − i Z ∞−∞ ˘ H ( x ) dx (cid:27) , (67)where ˘ U ( x, x ) is the evolution operator for the Hamiltonian ˘ H ( x ) and the initial ‘time’ x , i.e.,˘ U ( x, x ) = T exp (cid:26) − i Z xx ˘ H ( s ) ds (cid:27) . (68)We can also establish (67) using (63) and˘ U ( x, x ) = ˘ G ( x ) ˘ G ( x ) − , (69)which follows from (61) and (65).Now, consider the truncated potentials v j given by (74), and let ˘ M j and M j be the analogs ofthe transfer matrices ˘ M and M for these potentials. Then, we can use (64), (67), the semi-groupmultiplication rule for the evolution operators, and the vanishing of ˘ H ( x ) for all x ∈ R at which v ( x ) = 0 to establish ˘ M = ˘ M n ˘ M n − · · · ˘ M . (70)Furthermore, becauselim x → + ∞ v ( x ) = lim x →−∞ v n ( x ) = 0 , lim x →±∞ v j ( x ) = 0 for j ∈ { , , · · · , n − } , Eq. (55) implies that M = ˘ M e ϑ − i σ , M n = e − ϑ + i σ ˘ M n , M j = ˘ M j for j ∈ { , , · · · , n − } . Substituting these relations and (55) in (70), we arrive at the composition property of the transfermatrix M , namely M = M n M n − · · · M . (71)Therefore, ˘ M and M share the composition property of the well-known transfer materix M for theshort-range potentials.According to (67), ˘ H ( x ) is the Hamiltonian operator for an effective two-level quantum systemwhose S -matrix yields the transfer matrix ˘ M of v . Similarly to H ( x ), this operator is σ -pseudo-Hermitian whenever v is real-valued, and σ -pseudo-normal otherwise. It is also traceless whichimplies det ˘ M = 1. This equation together with (64) and det e ± ϑ ± i σ = 1 lead to another proof ofthe fact that det M = 1.The argument leading to (67) is clearly applicable to short-range potentials. For a short-rangepotential, S ( x ) = kx + ς ( x ) → kx + ϑ ± as x → ±∞ . According to (13), (26), (58), and (59), thisimplies ˘Ψ( x ) = e − iς ( x ) σ Ψ( x ) , (72)˘ U ( x, x ) = e − iς ( x ) σ U ( x, x ) e iς ( x ) σ , (73)where we have also employed (35) and (69). Taking x → + ∞ and x → −∞ in (73) and makinguse of (32) and (67) we recover (57). The main difference between H ( x ) and ˘ H ( x ) is that the latter is diagonalizable. Long-range potentials as short-range perturbations ofexactly solvable potentials
Consider a long-range potential v fulfilling (47). For every positive real number a of our choice, wecan dissect the real line into the intervals: I − := ( −∞ , − a ] , I := ( − a, a ) , I + := [ a, + ∞ ) , introduce the potentials v j ( x ) := (cid:26) v ( x ) for x ∈ I j , x / ∈ I j , (74)with j ∈ {− , , + } , so that v = v − + v + v + , (75)and express the transfer matrix ˘ M of v in the form ˘ M = ˘ M + ˘ M ˘ M − , where ˘ M j is the transfermatrix of v j . Clearly, v is a short-range potential. Therefore, in dealing with the difficultiesassociated with the long range of v , we can focus our attention to v ± . Because under a reflection(parity) transformation v − is mapped to a potential with the same structure as v + , we confine ourinvestigation to long-range potentials of the form v + , i.e., those supported in I + . In the following,we derive a decomposition of v + into the sum of a short-range potential u and an exactly solvablelong-range potential w . This is of interest, because for sufficiently large values of a , we can treat v + as a perturbation of w .Let ǫ be a real number such that 0 < ǫ <
1. Because v belongs to C α> / , for every k there is apositive real number a such that (cid:12)(cid:12)(cid:12)(cid:12) − v ( x )2 k (cid:12)(cid:12)(cid:12)(cid:12) ≥ ǫ for all x ≥ a . (76)In the following, we choose a ≥ a and introduce the functions f ± : [ a, ∞ ) → C and ψ ± : R → C according to f ± ( x ) := e ± iS ( x ) p − v ( x ) / k for x ≥ a, (77) ψ ± ( x ) := (cid:26) f ± ( x ) for x ≥ a, a ± e ikx + b ± e − ikx for x < a, (78)where S is given by (37), and a ± and b ± are complex coefficients that render ψ ± differentiable at x = a , i.e., a ± = e − ika (cid:2) f ± ( a ) − if ′± ( a ) /k (cid:3) , (79) b ± = e ika (cid:2) f ± ( a ) + if ′± ( a ) /k (cid:3) . (80)It is not difficult to check that ψ ± are solutions of the time-independent Schr¨odinger equation(2) for a potential of the form, w ( x ) := (cid:26) v ( x ) − u ( x ) for x ≥ a, x < a, (81)11here u ( x ) := k (cid:20) v ( x ) − v ′ ( x ) k τ ( x ) − v ′′ ( x ) τ ( x ) (cid:21) for x ≥ a, x < a, (82) τ ( x ) := 1 − v ( x )2 k . (83)Because ψ ± are linearly independent, every solution ψ of the Schr¨odinger equation (2) for thepotential w is a linear combination of ψ ± ; there are complex coefficients A and B such that ψ ( x ) = A ψ + ( x ) + B ψ − ( x ) . (84)We can use this relation to determine the transfer matrix of w . To this end, we first introduce : ς u ( x ) := − k Z x u ( s ) ds, S w ( x ) := S ( x ) − ς u ( x ) = kx − k Z x w ( s ) ds, (85)˘ A + := A e iς u ( ∞ ) , ˘ B + := B e − iς u ( ∞ ) , (86)and use (77), (78), (84), and (85) to show that ψ ( x ) → ˘ A + e iS w ( x ) + ˘ B + e − iS w ( x ) for x → + ∞ . (87)Moreover, for x < a , ψ ( x ) = ( a + A + + a − B + ) e ikx + ( b + A + + b − B + ) e − ikx = ( a + e − iς u ( ∞ ) ˘ A + + a − e iς u ( ∞ ) ˘ B + ) e iS w ( x ) +( b + e − iς u ( ∞ ) ˘ A + + b − e iς u ( ∞ ) ˘ B + ) e − iS w ( x ) , (88)where we have made use of (86) and the fact that S w ( x ) = kx for x < a .Next, we observe that because (88) holds for x → −∞ , we can identify the coefficients of e iS w ( x ) and e − iS w ( x ) on the right-hand side of (88) with ˘ A − and ˘ B − . This yields a pair of linear equationsfor ˘ A + and ˘ B + . Expressing the solution of these equations in the form (39), we find the followingformula for the transfer matrix ˘ M of the potential w , which we label by ˘ M w .˘ M w = (cid:20) a + e − iς u ( ∞ ) a − e iς u ( ∞ ) b + e − iς u ( ∞ ) b − e iς u ( ∞ ) (cid:21) − = h ˘ G w ( a ) e − iς u ( ∞ ) σ i − = e iς u ( ∞ ) σ ˘ G w ( a ) − . (89)where ˘ G w is the matrix-valued function (61) associated with the potential w , and we have employedthe identity, ˘ G w ( a ) = (cid:20) a + a − b + b − (cid:21) , (90)which we obtain by setting ψ = ψ + and ψ = ψ − in (60) and using the resulting equation togetherwith (61) and (78) to compute ˘ G w ( a ). Because ˘ M w has a unit determinant, (89) implies a + b − − a − b + = 1 . (91) Because u is a short-range potential, ς u ( ∞ ) := lim x →∞ ς u ( x ) exists.
12e can indeed verify this relation by exploiting the fact that the computation of the Wronskianof the solutions ψ ± at x = 0 and in the limit x → ∞ gives the same result. Employing (78) toperform this calculation, we respectively find 2 ik ( a + b − − a − b + ) and 2 ik . Hence (91) holds. Usingthis equation in Eqs. (89), we have˘ M w = (cid:20) b − e iς u ( ∞ ) − a − e iς u ( ∞ ) − b + e − iς u ( ∞ ) a + e − iς u ( ∞ ) (cid:21) . (92)Next, we recall that, in light of (74) and (81), v + ( x ) = (cid:26) u ( x ) + w ( x ) for x ≥ a, x < a. (93)If there is some α > / v ( x ) ∝ x − α as x → + ∞ , then (82) implies that in this limit u ( x ) ∝ x − α ′ for some α ′ >
1, i.e., u is a short-range potential. According to (82), this is generallytrue, for potentials v of class C α> / such that v ′ also belongs to C α> / and v ′′ is a short-rangepotential. Under these conditions v + is the sum of a short-range potential u and an exactly solvablelong-range potential w .The constructions leading to (93) are clearly valid for every a ≥ a . This together with thefact that for larger values of a we can treat u as a small perturbation of w suggest using first-orderperturbation theory to compute the transfer matrix ˘ M of v + , which we denote by ˘ M + .Let Ψ q , H q ( x ), and U q ( x, x ) respectively stand for the two-component wave function (26),the Hamiltonian (28), and the evolution operator (31) for the potential q ∈ { v + , w, u } . Then, thetwo-component wave function Φ defined by,Φ( x ) := U w ( x, a ) − Ψ v + ( x ) , (94)satisfies i Φ ′ ( x ) = H ( x )Φ( x ) for H ( x ) := U w ( x, a ) − H u ( x ) U w ( x, a ) (95)= u ( x )2 k U w ( x, a ) − e − ikx σ K e ikx σ U w ( x, a ) . In other words, Φ( x ) = U ( x, x )Φ( x ) , (96)where U ( x, x ) := T exp (cid:26) − i Z xx H ( s ) ds (cid:27) . (97)If we respectively denote the two-component wave function (59), the Hamiltonian (66), and theevolution operator (68) for the potential q by ˘Ψ q , ˘ H q ( x ), and ˘ U q ( x, x ), with the help of (72) and(73), we can express (94) in the formΦ( x ) = ˘ U w ( x, a ) − e iς u ( x ) σ ˘ U v + ( x, x ) ˘Ψ v + ( x ) , (98)where ς q ( x ) := − k R x q ( s ) ds , and we have benefitted from the identities: ς w ( a ) = 0 and ς v + − ς w = ς u .For x = x , (98) gives Φ( x ) = ˘ U w ( x , a ) − e iς u ( x ) σ ˘Ψ v + ( x ). Solving this equation for ˘Ψ v + ( x ) andinserting the result in (98), we recover (96) with U ( x, x ) = ˘ U w ( x, a ) − e iς u ( x ) σ ˘ U v + ( x, x ) e − iς u ( x ) σ ˘ U w ( x , a ) . U v + ( x, x ) = e − iς u ( x ) σ ˘ U w ( x, a ) U ( x, x ) ˘ U w ( x , a ) − e iς u ( x ) σ . (99)Next, we recall that ς u ( a ) = 0 and the transfer matrices of v + and w are respectively given by˘ M + = ˘ U v + ( ∞ , −∞ ) = ˘ U v + ( ∞ , a ) , ˘ M w = ˘ U w ( ∞ , −∞ ) = ˘ U w ( ∞ , a ) . (100)In view of these observations, letting x = a and x → ∞ in (99) and making use of (89), we arriveat ˘ M + = e − iς u ( ∞ ) σ ˘ M w U ( ∞ , a ) = ˘ G w ( a ) − U ( ∞ , a ) . (101)Given that we have an explicit formula for ˘ G w ( a ), namely (90), this equation reduces the solutionof the scattering problem for v + to that of the determination of U ( ∞ , a ). Because u is a short-range potential, for sufficiently large values of a , we can find positive numbers γ and δ such that | u ( x ) | ≤ γk − δ x − (1+ δ ) for x ≥ a . Hence, Z ∞ a | u ( x ) | dx ≤ γkδ ( ak ) δ . (102)This relation together with the expression (95) for the Hamiltonian H ( x ) and the Dyson seriesexpansion of U ( ∞ , a ), i.e., U ( ∞ , a ) = I + ∞ X ℓ =1 ( − i ) ℓ Z ∞ a dx ℓ Z x ℓ a dx ℓ − · · · Z x a dx H ( x ℓ ) H ( x ℓ − ) · · · H ( x ) , (103)suggest the possibility of devising a perturbative method of computing U ( ∞ , a ) that involves thetruncation of its Dyson series. Retaining the first n + 1 terms of this series, we obtain an n -th orderperturbative expression for U ( ∞ , a ) with ( ak ) − δ playing the role of the perturbation parameter.Consider the fundamental matrix, F w ( x ) := (cid:20) ψ + ( x ) ψ − ( x ) ψ ′ + ( x ) ψ ′− ( x ) (cid:21) , where ψ ± are the solutions (78)of the Schr¨odinger equation (2) for the potential w . Then according to (61), (69), (73), and (78),for all x > a , U w ( x, a ) = e iς w ( x ) σ ˘ G w ( x ) ˘ G w ( a ) − , (104)˘ G w ( x ) = e − iS ( x ) σ G ( x ) e iS ( x ) σ , (105)where G ( x ) := µ + ( x ) I + µ − ( x ) σ + iν ( x ) K , (106) µ ± ( x ) := 1 ± τ ( x )2 p τ ( x ) , ν ( x ) := − v ′ ( x )8 k p τ ( x ) , (107) τ is the function defined by (83), and we have made use of the fact that ς w ( a ) = 0. Because theimaginary part of v + is a short-range potential, e ± iS ( x ) , e ± iς w ( x ) , and consequently the entries of14 G w ( x ) and U w ( x, a ) are bounded functions of x . By virtue of (28) and (95), this implies that theentries of H u ( x ) and consequently H ( x ) are products of u ( x ) and certain bounded functions of x .Therefore, there is a positive real number β such that the entries H ij ( x ) of H ( x ) satisfy | H ij ( x ) | ≤ βk − | u ( x ) | . (108)Now, let U ( ℓ ) denote the ℓ -th term in the Dyson series expansion (103) of U ( ∞ , a ), i.e., U ( ℓ ) := Z ∞ a dx ℓ Z x ℓ a dx ℓ − · · · Z x a dx [ H ( x ℓ ) H ( x ℓ − ) · · · H ( x )] , (109)and U ( ℓ ) ij label its entries. Then, in view of (102), (108), and the fact that the entries of the matrix H ( x ℓ ) H ( x ℓ − ) · · · H ( x ) have the form P k =1 P k =1 · · · P k ℓ =1 H ik H k k · · · H k ℓ j , we infer | U ( ℓ ) ij | ≤ (cid:18) βk (cid:19) ℓ Z ∞ a dx ℓ Z x ℓ a dx ℓ − · · · Z x a dx | u ( x ℓ ) u ( x ℓ − ) · · · u ( x ) |≤ ℓ ! (cid:18) βk (cid:19) ℓ Z ∞ a dx ℓ Z ∞ a dx ℓ − · · · Z ∞ a dx | u ( x ℓ ) u ( x ℓ − ) · · · u ( x ) |≤ ℓ ! (cid:20) βγδ ( ka ) δ (cid:21) ℓ . (110)This shows that the error associated with the approximation, U ( ∞ , a ) ≈ I + n X ℓ =1 U ( ℓ ) , (111)is proportional to ( ka ) − nδ . Hence, we can reduce it by adopting larger values of a .Let us examine the first-order approximation. Substituting (104) in (95) and making use of(109) and (111) with n = 1, we find H ( x ) = ˘ G w ( a ) H ( x ) ˘ G w ( a ) − , (112) U ( ∞ , a ) ≈ I + ˘ G w ( a ) Z ∞ a H ( x ) dx ˘ G w ( a ) − , (113)where H ( x ) := u ( x )2 k G u ( x ) − K G u ( x ) , G u ( x ) := e − iς u ( x ) σ G ( x ) e iS ( x ) σ . Now, consider setting n = 1 in (111). This amounts to ignoring quadratic and higher order termsin powers of ( ak ) − δ . With the help of (102), we observe that | ς u ( x ) | ≤ k Z xa | u ( s ) | ds ≤ k Z ∞ a | u ( s ) | ds ≤ γδ ( ak ) δ . Because v and v ′ belong to C α> / , µ ± and ν are bounded functions for x > a and a sufficiently large. u ( x ) e ± iς u ( x ) σ , we can approxi-mate these terms by u ( x ) I . In view of this observation and the identities, K σ = − σ K = σ and K = , (113) gives U ( ∞ , a ) ≈ I + ˘ G w ( a ) U ˘ G w ( a ) − , (114)where U := 12 k Z ∞ a u ( x ) τ ( x ) e − iS ( x ) σ K e iS ( x ) σ dx = (cid:20) U U − −U + −U (cid:21) , (115) U := 12 k Z ∞ a u ( x ) τ ( x ) dx, U ± := 12 k Z ∞ a u ( x ) e ± iS ( x ) τ ( x ) dx. (116)Because u and Im( v ) belong to C α> , and S ( x ) = kx + ς ( x ), the improper integrals yielding U and U ± converge.Substituting (114) in (101), we obtain the following approximate expression for the transfermatrix of v + . ˘ M + ≈ ( I + U ) ˘ G w ( a ) − . (117)In view of (89) and the fact that e − iς u ( ∞ ) σ − I contributes as a first-order term in our perturbationscheme, we can also express (117) in the form, ˘ M + ≈ ˘ M (0)+ + ˘ M (1)+ , where˘ M (0)+ := ˘ M w , ˘ M (1)+ := [ U − iς u ( ∞ ) σ ] ˘ M w . Clearly, ˘ M + ≈ ˘ M (0)+ gives the zeroth-order approximation corresponding to U ( ∞ , a ) ≈ I .As an example consider the potential, v ( x ) = gx + z x , (118)where g and z are respectively real and complex coupling constants. It clearly satisfies (47). There-fore, whenever a | g | + | z | < ak ) and ak ≫
1, we can use the above perturbation scheme todetermine the transfer matrix ˘ M + and the reflection and transmission amplitudes, R l/r and T , ofthe potential: v + ( x ) = gx + z x for x ≥ a, x < a. (119)Fig. 1 shows the graphs of the reflection and transmission coefficients, |R l/r | and |T | , of thispotential for g = − /a , z = 5 − i , and ak ≥
5. The dashed and solid curves correspond to theresults of the zeroth- and first-order perturbative calculations, respectively. As expected, theirdifference diminishes as ak grows.Let us recall the decomposition (75) of the potential v . We can use the above perturbativeapproximation scheme to compute the transfer matrix ˘ M − of the potential v − and use the compo-sition rule for transfer matrices to determine the transfer matrix ˘ M of the potential v . A desirableaspect of this approximation scheme is that we can improve its accuracy not only by includinghigher order terms in the perturbative expansion of ˘ M ± , but also by choosing larger values of a which would reduce the perturbation parameter ( ak ) − . Indeed for every value of k , we can adjust a so that ak attains such a large value that even the zeroth-order approximation is valid. Thismarks an important distinction between our scheme and the standard WKB approximation which16 | ℛ l | ℛ r | Figure 1: Plots of |R l | , |R r | , and |T | as functions of ak for the potential (119) with g = − a − and z = 5 − i . The dashed (dark blue) and solid (purple) curves respectively correspond to theresults of the zeroth- and first-order perturbative calculations. They converge for larger values of ak .is generically valid for high energies. The price one pays for taking large values of a is to increasethe size of the support of the potential v . This can in principle complicate the computation of itstransfer matrix. But, v has a finite range, and there are well-known numerical methods for a director indirect determination of its transfer matrix. A basic property of second order linear homogeneous ordinary differential equations is that givena nonzero solution of this equation we can obtain a second linearly independent solution [34]. Inthis section, we use this property to generate an exactly solvable long-range potential of the form(119). Finding an exact expression for the general solution of the Schr¨odinger equation (2) for thispotential is equivalent to the exact solution of (2) for the potential v in the interval [ a, ∞ ).Let φ + : [ a, ∞ ) → C be a function of the form, φ + ( x ) := e ξ ( x ) e iS v + ( x ) , (120)where ξ : [ a, ∞ ) → C is an axillary function, S v + ( x ) := kx + ς + ( x ), and ς + ( x ) := − k Z x v + ( s ) ds = − k Z xa v ( s ) ds. (121)Demanding φ + to solve the Schr¨odinger equation, − ψ ′′ ( x ) + [ v ( x ) + q ( x )] ψ ( x ) = k ψ ( x ) , (122)in the half-line [ a, ∞ ), we find q = ξ ′′ + ξ ′ + 2 ik (cid:16) − v k (cid:17) ξ ′ − v k − iv ′ k . (123)17 + is a solution of the Schr¨odinger equation (2) for the potential v provided that we select ξ insuch a way that q = 0. For a potential of the form (118), we can satisfy this equation using theansatz ξ ( x ) = c x , (124)where c is a constant. Substituting (118) and (124) in (123) and demanding its right-hand side tovanish, we obtain c = c ⋆ := g (2 k + ig )8 k , (125) z = z ⋆ := 2 ik c = g ( − g + 2 ik )4 k . (126)These in turn imply S v + ( x ) = kx − k (cid:20) g ln (cid:16) xa (cid:17) + z ( x − a ) ax (cid:21) , (127) φ + ( x ) = e c ⋆ /a exp { i [ kx − ( g/ k ) ln( x/a )] } . (128)The latter is an exact solution of the Schr¨odinger equation (2) for the potential (118) in [ a, ∞ )provided that (125) holds. According to (120) and (124), φ + has the appealing property: φ + ( x ) → e iS v + ( x ) for x → ∞ . (129)In Appendix we construct another solution, φ − , of the same Schr¨odinger equation that satisfies φ − ( x ) → e − iS v + ( x ) for x → ∞ . (130)It is given by φ − ( x ) := 1 φ + ( x ) + igk φ + ( x ) Z ∞ x dss φ + ( s ) . (131)Now, consider the potential (119) with z given by (126), i.e., v + ( x ) = gx + z ⋆ x for x ≥ a, x < a. (132)The above analysis shows that the corresponding Schr¨odinger equation (2) admits a pair of linearlyindependent solutions of the form (78) with f ± replaced with φ ± . We can follow the approach ofSec. 4 to express the transfer matrix ˘ M for this potential in the form,˘ M + = (cid:20) b − − a − − b + a + (cid:21) , (133)where a ± and b ± are given by (79) and (80) with f ± replaced with φ ± . In view of (126), (128), and(131), this gives a − = (cid:2) − b g e − iak + 2 i b g (1 − b g ) I (cid:3) e − c /a , a + = (1 − b g ) e c /a , (134) b − = (cid:0) b g + 2 i b g e iak I (cid:1) e − c /a , b + = b g e iak e c /a , (135)18 ak | ℛ l (cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17) ak | ℛ r (cid:18) (cid:19)(cid:20)(cid:21) ak | Figure 2: Plots of |R l | , |R r | , and |T | as functions of ak for the potential (119) with g = − a − and z given by (126). The dashed (dark blue) and solid (red) curves respectively correspond tothe results of the zeroth-order perturbative calculations of Sec. 5 and the exact results given byEqs. (137), (138), and (139).where b g := g/ ak , I := 2 ak e c /a Z ∞ a dxx φ + ( x ) = (2 ak ) − ig/k Z ∞ ak t − ig/k e − it dt = e πg/ k (2 ak ) − ig/k Γ( ig/k, iak ) , (136)and Γ( · , · ) stands for the incomplete Gamma function [47].Having obtained the transfer matrix ˘ M + , we use (56) to calculate the reflection and transmissionamplitudes, R l/r and T , of the potential (132). Because for this potential ϑ − i = 0 and ϑ + i = − Im( z ) / ak = − b g , this yields R l = b + a + = b g e iak − b g , (137) R r = − e b g a − a + = e − iak b g (cid:18) b g e − iak − b g − i b g I (cid:19) , (138) T = e b g a + = e − iak b g − b g . (139)Fig. 2 shows plots of the reflection and transmission coefficients for the potential (132) with g = − a − . For large values of ak the approximate results obtained using the zeroth-order perturbationschemes of Sec. 4 are in perfect agreement with the exact results provided by (137) – (139). Transfer matrices have been extensively used in dealing with scattering problems since the 1940’s.Their applications were however confined to the study of short-range potentials. In this article, weextended their domain of application to a large class of real and complex long-range potentials. This19nvolved a re-examination of the standard notion of the transfer matrix of a short-range potential M ,its identification with the S-matrix of a certain effective two-level quantum system, its relationshipwith the classical notion of the fundamental matrix of the theory of linear ordinary differentialequations, and most notably the introduction of a pair of transfer matrices ˘ M and M for lang-range potentials which shared the basic features of M . In particular, they store the informationabout the scattering features of the potential and possess the same composition property.We have employed the composition property of ˘ M to reduce the problem of dealing with thelong-range potentials of our interest to those supported in an interval of the form [ a, ∞ ) with a > M whose accuracy can be improved by choosinglarger values of a .In order to demonstrate the utility of our approximation scheme, we have introduced an exactlysolvable long-range complex potential and compared the outcome of the exact and approximatecalculations of its reflection and transmission coefficients. Our explicit calculations reveal an almostperfect agreement between the exact and approximate results for ak ≫ Appendix: Construction of φ − Consider the solution φ + of the Schr¨odinger equation (2) for the potential (118) with z given by(126). We can use φ + to express the general solution of (2) in [ a, ∞ ) as [34], ψ = c + φ + + c − ψ − , (140)where c ± are constant coefficients, ψ − ( x ) := φ + ( x ) Z xb dsφ + ( s ) . (141)and b is a real number exceeding a . In view of (120), ψ − ( x ) = φ + ( x ) Z xb e − ξ ( s )+ iς + ( s )] e − iks ds = φ + ( x ) (cid:20) i k φ + ( s ) (cid:12)(cid:12)(cid:12)(cid:12) xb + ik Z xb ξ ′ ( s ) + iς ′ + ( s ) φ + ( s ) ds (cid:21) = i k (cid:20) φ + ( x ) − φ + ( x ) φ + ( b ) + igk φ + ( x ) Z bx dss φ + ( s ) (cid:21) , where we have performed an integration by parts and employed (118), (121), (124), and (126). Now,let c − = − ik and c + = φ + ( b ) − . Then for each b > a , (140) produces the following solution of theSchr¨odinger equation (2) for the potential (118) in [ a, ∞ ). ψ b ( x ) := 1 φ + ( x ) + igk φ + ( x ) Z bx dss φ + ( s ) . With the help of (128) we can show that R bx dss φ + ( s ) = h [ I (2 kb ) −I (2 kx )], where h := e − c /a (2 ak ) − ig/k , I ( y ) := R y t − ig/k e − it dt = e πg/ k Γ( ig/k, i, iy ), and Γ( · , · , · ) is the generalized incomplete Gamma20unction. It is not difficult to see that for y ≥ |I ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) y − ig/k e − iy − e − i + (cid:18) − igk (cid:19) Z y t − ig/k e − it dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ y + 1 + (cid:18) g k (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)Z y t − ig/k e − it dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ y + 1 + (cid:18) g k (cid:19) Z y dtt = 2 + g k (cid:18) − y (cid:19) . (142)where we have used integration by parts in the first line and benefitted from the fact that g is real.According to (142), I ( ∞ ) := lim y →∞ I ( y ) exists and the improper integral R ∞ x dss φ + ( s ) converges.This in turn implies that the φ − given by (131) is a well-defined function in the interval [ a, ∞ ). Itis easy to check that it solves the Schr¨odinger equation (2) for the potential (118) with z given by(126) and that it satisfies (129). Acknowledgements
We thank Turkish Academy of Sciences (T ¨UBA) for supporting FL’s visit to Ko¸c University in 2019during which this work was initiated. AM has been supported by T ¨UBA’s membership grant.
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