Analyticity properties of the scattering matrix for matrix Schrödinger operators on the discrete line
Miguel Ballesteros, Gerardo Franco Córdova, Hermann Schulz-Baldes
AAnalyticity properties of the scattering matrixfor matrix Schr¨odinger operators on the discrete line
Miguel Ballesteros , Gerardo Franco C´ordova , Hermann Schulz-Baldes IIMAS, UNAM, Cuidad de Mexico, Mexico Department Mathematik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Germany
Abstract
Explicit formulas for the analytic extensions of the scattering matrix and the timedelay of a quasi-one-dimensional discrete Schr¨odinger operator with a potential of finitesupport are derived. This includes a careful analysis of the band edge singularities andallows to prove a Levinson-type theorem. The main algebraic tool are the plane wavetransfer matrices. MSC2010 database: 47A40, 81U05, 47B36
This paper addresses the stationary scattering theory of self-adjoint matrix Jacobi operators H on (cid:96) ( Z , C L ) of the form( H u )( n ) = u ( n + 1) + V ( n ) u ( n ) + u ( n − , n ∈ Z , (1)where V ( n ) = V ( n ) ∗ ∈ C L × L is a selfadjoint L × L matrix and L ∈ N is a fixed number.Such tridiagonal operators are also called Jacobi operators and have a tight connection toorthogonal (matrix) polynomials and via their spectral theory to (matrix-valued) measureson the real line. They are the discrete analogues of Sturm-Liouville operators and many ofthe techniques such as oscillation theory and Weyl extension theory transpose after suitablemodifications. Moreover, these operators are widely used for modeling low-energy phenomenain solid state physics in a one-particle framework. Here the focus will be on an elementaryscattering situation where V ( n ) is non-vanishing only for a finite number of sites n , namelythe perturbation V of the discrete Laplacian H = H − V has finite support. Most standardworks [19, 28, 27, 26] cover this situation and thus lead to numerous general results. Morerecently, the case of non-linear potentials has also been addressed, see [9, 18] and referencestherein. Furthermore, the scalar case L = 1 has been treated in detail by Hinton, Klaus andShaw a long time ago [10], see also the recent contribution [11]. For continuous matrix-valuedSchr¨odinger operators there are works by Klaus [14] and Aktosun, Klaus and Van Der Mee1 a r X i v : . [ m a t h - ph ] A p r
1] as well as a more recent contributions by Aktosun, Klaus and Weder [4, 2, 5]. Also thehalf-space version of the above discrete matrix Schr¨odinger equation has been analysed [3, 20].In spite of all these prior works, this paper has a several novel facts and features that solidifyour understanding of stationary scattering theory: • The analytic structure of the scattering matrix and the time delay in complex energy E or rather the parameter z defined by E = z + z − is studied in detail. • The unitarity relation of the scattering matrix is extended to complex energies, includingthe band edge thresholds. • The analyticity, J -unitarity and multiplicativity of the newly introduced plane wavetransfer matrices are consistently used in the arguments, leading to elementary algebraicproofs of the main results. • The connections between the plane wave transfer matrices and the standard transfermatrices used in the theory of Jacobi operators is established. • As an application, a Levinson-like theorem connecting the total time delay to the numberof bound and half-bound states is proved using purely complex analytic means (namely,the argument principle). Strictly speaking this is a new result, even though there arenumerous prior contributions (in particular, [4]).Let us begin by recalling the elementary spectral analysis of H . By Weyl’s theorem, theessential spectrum of H is given by the (absolutely continuous) spectrum σ ( H ) = [ − , H . It will be part of the results below (Theorem 9) that the remainder of the spectrumof H only consists of a finite number J b of eigenvalues E , . . . , E J b ∈ R (listed with theirmultiplicity) outside of [ − , σ p ( H ) = { E , . . . , E J b } of H , notably there are no embeddedeigenvalues in the present situation. At the band edge thresholds E = ±
2, there may be anumber J h of further so-called half-bound states that will be discussed as well.For the analysis of the scattering problem and the analytic continuation of the scatteringmatrix, it is convenient to study the Jacobi equation Hu = Eu for complex energies E andmatrix-valued or vector-valued functions of the form u : Z → C L × L or u : Z → C L where H is given by the same prescription as in (1). The Jost solutions are particular such solutionswhich are fixed by their asymptotic behavior at ±∞ dictated by solutions of the free Jacobiequation H u = Eu associated to the free Hamiltonian H . For any E ∈ C \ {− , } thereare two such free solutions u z ( n ) = z n , u /z ( n ) = z − n , where ∈ C L × L the identity matrix and z, z − ∈ C are the two solutions of E = z + z − . (2)Note that the map z ∈ C \ S (cid:55)→ E ∈ \ [ − ,
2] is two-to-one and the associated Riemannsurface for E consists of two sheet that are connected through branching on [ − , u z , u /z ) is a fundamental solution of H u = Eu . Let us point out that for E (cid:54)∈ ( − , u z are exponentially increasing or decreasing in n , while for z ∈ S \ {− , } they are the well-known plane wave solutions. Further note that for z = ± E = ±
2, there is only one free solutions u ± . There is, however, again asecond solution v ± of H u = ± u given by v ± ( n ) = ( ± n n . This second solution is henceincreasing linearly in n . Now the Jost solutions are defined. Definition 1.
The Jost solutions u z ± : Z → C L × L of Hu = Eu with E = z + z − are definedby the equalities u z + ( n ) = u z ( n ) , u z − ( − n ) = u z ( − n ) , holding for large enough n ∈ N such that the support of V is contained in ( − n, n ) . For | z | < , the Jost solution u z + is called right-decreasing and u z − left-increasing, while u /z + iscalled right-increasing and u /z − left-decreasing. There are in total 2 L linearly independent solutions u : Z → C L of the equation Hu = Eu .For E ∈ C \ {− , } , namely z ∈ C \ {− , , } , the columns of the matrix ( u z + , u /z + ) providea basis of the space of solutions. The same holds for and ( u z − , u /z − ). One can thus expandany solution w.r.t. one of these basis. This justifies that the following definition is possible. Definition 2.
For z ∈ C \ {− , , } , the plane wave transfer matrix M z ∈ C L × L is definedby (cid:0) u z − , u /z − (cid:1) = (cid:0) u z + , u /z + (cid:1) M z . (3) The L × L entries of the transfer matrix are denoted by M z = (cid:32) M /z − N z − N /z − M z − (cid:33) . (4)Note that (3) contains the equations u z − = u z + M /z − + u /z + N /z − and u /z − = u z + N z − + u /z + M z − ,which are actually identical if z is replaced by z − . Provided that M z − is invertible, one canrewrite the second equation as u /z + = u /z − ( M z − ) − − u z + N z − ( M z − ) − . In a similar manner, onecan expand u z + in terms of u /z + and u z − , provided that M /z − is also invertible. The scatteringmatrix is then defined by a relation similar to (3). To shorten notations in the following, letus introduce the set C of those points in z ∈ C \ {− , , } at which both M /z − and M z − areinvertible. It will be shown (Corollary 19 below) that all points in the closed unit disc are in C , except those z ∈ ( − ,
1) for which z + z − is an eigenvalue of H . Definition 3.
For any z ∈ C , the scattering matrix S z ∈ C L × L expresses the Jost solutions u z − and u /z + in terms of u z + and u /z − , namely it is defined by (cid:0) u z − , u /z + (cid:1) = (cid:0) u z + , u /z − (cid:1) S z . (5) The L × L entries of the scattering matrix are denoted by S z = (cid:18) T z + R z − R z + T z − (cid:19) , and are called the transmission coefficients T z ± and reflection coefficients R z ± . z u − + +z u − + − u φφψψ −+ Figure 1: Schematic representation of the scattering process for | z | <
1. The arrows indicatethe direction in which the solutions decrease.Of course, there are tight connections between the coefficient matrices of M z and S z . Forexample, the relation u /z + = u /z − ( M z − ) − − u z + N z − ( M z − ) − noted above directly implies that T z − = ( M z − ) − and R z − = − N z − ( M z − ) − . Let us next provide a list of basic algebraic propertiesof the plane wave transfer matrix and scattering matrix. Some will be expressed in terms ofthe real Pauli matrices of size 2 L × L with L × L blocks: K = (cid:18) (cid:19) , I = (cid:18) − (cid:19) , J = (cid:18) − (cid:19) . (6)Furthermore, let M ∗ denote the adjoint (conjugate transpose) of a matrix M and ı = √− Proposition 4. (i)
The plane wave transfer matrix satisfies for all z ∈ C \ {− , , } ( M /z ) ∗ J M z = J , M z J ( M /z ) ∗ = J . (7) For z ∈ S \ {− , } , this shows that M z is J -unitarity, namely ( M z ) ∗ J M z = J . (ii) For z ∈ R , the plane wave transfer matrix is I -unitary, namely ( M z ) ∗ IM z = I . (iii) For all z ∈ C , one has ( S /z ) ∗ S z = , S z = K ( S z ) ∗ K . For z ∈ S \ {− , } , this implies that S z is unitary, ( T z + ) ∗ = T z − and ( R z ± ) ∗ = R z ± . (iv) If V = 0 , one has M z = and S z = for all z ∈ C . (v) Both M z and S z are meromorphic on C . The proofs of (i) and (ii) are given Section 4, and that of (iii) and (v) in Section 6. Item(iv) is obvious. A detailed analysis of the singularities and zeros of M z and S z and theirmatrix entries make up a large part of the paper, namely Sections 13 through 10. Beforecoming to the main results of the paper, let us make several remarks on structural propertiesof the plane wave transfer matrices and the scattering matrix. Remark 5.
There are two main reasons justifying the terminology plane wave transfer matrix for M z . First of all, it reflects that the plane wave solutions from the right are transferred4o the plane wave solutions on the left, see Figure 1. The second reason is that M z is theordered product of single site plane wave transfer matrices defined by M z ( n ) = (cid:18) (cid:19) + ıν z (cid:18) V ( n ) z − n V ( n ) − z n V ( n ) − V ( n ) (cid:19) , (8)where ν z = ız − z − (9)is a quantity that will be appear in many formulas below. It satisfies ν z = − ν /z = ( ν /z ) ∗ ,is analytic in C \ {− , , } and has poles of first order at ±
1. Then the multiplicativity, adefining property of any type of transfer matrices, is given by M z = M z ( K + ) M z ( K + − · · · M z ( K − + 1) , where [ K − + 1 , K + ] ∩ Z is the support of V . The multiplicativity will be explained in detailin Section 4 and exploited later on. As far as we know, one novel point of this paper is howto obtain M z and M z ( n ) from the standard transfer matrix used in the theory of (block)Jacobi matrices, see Section 4. (cid:5) Remark 6.
In this remark, z ∈ S . Then there are two perspectives on the passage from M z to S z . The first one is via the singular value decomposition and is standard in thephysics literature. It goes back at least to [17], and is, from a mathematical perspective, alsodeveloped in the appendix of [22] so that no further details are given here. The second one isbased on the fact that the graph of J -unitary matrix is Lagrangian w.r.t. J ⊕ J and is nowdescribed in some detail. Let us first recall (in slightly different variant than in [25, 24, 16])that to any J -unitary M (namely, satisfying M ∗ J M = J so that also MJ M ∗ = J ), thereis an associated unitary V ( M ) ∈ C L × L by V ( M ) = (cid:18) ( A ∗ ) − − BD − D − C D − (cid:19) = (cid:18) ( A ∗ ) − − BD − B ∗ ( A ∗ ) − D − (cid:19) , M = (cid:18) A BC D (cid:19) . (10)This unitary describes geometrically the twisted graph of M , see [25]. It satisfies V ( M ) = V ( M − ) ∗ = J V ( M ∗ ) ∗ J , and, moreover, there is a tight connection between the spectral theory of M and V ( M ),namely the multiplicity of 1 as eigenvalue of M is equal to the multiplicity of 1 as eigenvalueof V ( M ). Furthermore, V is a bijection from the set of J -unitaries onto the set of elementsin the unitary group U(2 L ) with invertible diagonal entries. In the present situation where M = M z for z ∈ S \ {− , , } , one finds explicitly that V ( M z ) = S z . (11)Hence the scattering matrix encodes the twisted graph of the transfer matrix M z . (cid:5) emark 7. In Definition 3, the scattering matrix acts from the right on the Jost solutions.In the solid state physics literature of quasi-one-dimensional systems, the scattering matrixusually acts from the left on vectors. The connections between these two points of view isestablished by selecting two particular solutions u z − ψ + and u /z + φ − by picking two vectors ψ + , φ − ∈ C L . Then the left-decreasing and right-increasing solutions u /z − ψ − and u z + φ + aregiven by vectors ψ − , φ + ∈ C L that are, according to (5), specified by (cid:0) u z − , u /z + (cid:1) (cid:18) ψ + φ − (cid:19) = (cid:0) u z + , u /z − (cid:1) S z (cid:18) ψ + φ − (cid:19) = (cid:0) u z + , u /z − (cid:1) (cid:18) φ + ψ − (cid:19) , namely S z (cid:18) ψ + φ − (cid:19) = (cid:18) φ + ψ − (cid:19) . Then there is a standard passage ( e.g. [17], see Figure 1) from the scattering matrix to thetransfer matrix M z sending the components ψ ± on the left to those ψ ± on the right: S z (cid:18) ψ + φ − (cid:19) = (cid:18) φ + ψ − (cid:19) ⇐⇒ M z (cid:18) ψ + − ψ − (cid:19) = (cid:18) φ + − φ − (cid:19) . Hence also the plane wave transfer matrix acts from the left on the coefficients of solutions.As already stated, the link to the standard transfer matrices is explained in Section 4. (cid:5)
Up to now, all the above follows from relatively standard techniques. The second mainpoint of this paper is the asymptotics of the scattering matrix in the band edges E = ± z = ±
1. It will be shown that the limits lim z →± S z always exists so thatthe singularity is removable (see Section 9, in particular, Proposition 24). Theorem 8.
The scattering matrix extends analytically to {− , } . Furthermore, explicit formulas for the above limits will be provided. For sake of notationalsimplicity, we will focus on z = 1, as z = − z → S z = (cid:18) (cid:19) . (12)There are, however, non-generic so-called exceptional cases for which the limits of the trans-mission matrices are also non-trivial and can be expressed explicitly in terms of Wronskiansof the Jost functions, see Section 9 below.The second main result concerns the Levinson Theorem in the present context. Suchtheorems connect the number of bound and half-bound states to the winding of the scatteringmatrix (also called the total phase shift). For discrete Schr¨odinger operators such a connectionhas been found for one-dimensional operators [10], quasi-one-dimensional ones on a half-line[4] and higher dimensional lattice operators [7]. It has also been understood that the equalityis of topological origin [13]. We will set S E = S z if (2) holds and (cid:61) m ( z ) > S z is easily expressed in terms of S z due to the relation S z = K ( S z ) ∗ K . The following result is proved in the Section 12.6 heorem 9. The Hamiltonian H only has a finite number J b of eigenvalues E , . . . , E J b ∈ R (listed with their multiplicity) and they are outside of [ − , . Moreover, at the thresholds E = ± , there are J ± h ≤ L bounded solutions of Hu = ± u which are called half-bound states.With J h = J − h + J + h , one has J b + J h − L = (cid:90) − dE πı Tr (cid:0) ( S E ) ∗ ∂ E S E (cid:1) . The third and last result worth mentioning in this introduction is a formula for the scat-tering matrix in terms of the Green function. One such connection is well-known and based ona formula for the wave operators (abelian limits) [28], but here another formula is presented.It uses the tight connection between transfer matrices and Green functions for finite Jacobimatrices, also in the matrix-valued case ( e.g. [23]). This is more in the spirit of applications inmesoscopic physics [6], but the formula below provides the full (complex) energy dependenceand hence goes beyond prior results. For n, m ∈ Z and E (cid:54)∈ σ ( H ), the Green function isdefined by G E ( n, m ) = π n ( H − E ) − ( π m ) ∗ ∈ C L × L , (13)where π n : (cid:96) ( Z , C L ) → C L is the partial isometry defined by π n φ = φ n ∈ C L for a vector φ = ( φ n ) n ∈ Z ∈ (cid:96) ( Z , C L ). The following formula is derived in Section 13, see equations (53)and (54). Proposition 10.
For | z | < with z + z − / ∈ σ ( H ) , the scattering matrix S z is given by S z = ( z − z − ) z K − − K + (cid:18) G E ( K + , K − ) z − K + − K − ( G E ( K + , K + ) + ıν z ) − z K + + K − ( G E ( K − , K − ) + ıν z ) G E ( K − , K + ) (cid:19) . In a follow-up to this paper, we will consider be the scattering situation with an unboundedsupport of V , but under a short range condition. Once the existence of Jost solutions is assuredby a standard argument based on the Volterra equation, many (but not all) of the results ofthis paper transpose to this more general situation. As already stressed, the support of the potential is supposed to be finite. For sake of con-creteness, let us suppose that the support is contained in {− K − + 1 , . . . , K + } for some finiteand fixed K ± with K − ≤ K + . It is simple to construct the Jost solutions explicitly by usingthe transfer matrices T E ( n ) = (cid:18) E − V ( n ) − (cid:19) . The transfer matrices over several sites are defined to be T E ( n, m ) = T E ( n ) · · · T E ( m + 1) , n > m , T E ( n, n ) = L . Also set T E ( m, n ) = T E ( n, m ) − . Then onehas T E ( n, n −
1) = T E ( n ) and the concatenation relation T E ( n, m ) T E ( m, k ) = T E ( n, k ).Moreover, T E ( n, m ) ∗ I T E ( n, m ) = I , (14)where I is second of the real Pauli matrices defined in (6). For real energies E ∈ R , thisstates that the transfer matrices lie in the group I -unitary matrices satisfying T ∗ IT = I .The transfer matrices allow to rewrite the Sch¨odinger equation Hu = Eu by settingΦ( n ) = (cid:18) u ( n + 1) u ( n ) (cid:19) ∈ C L × L , (15)namely one has Φ( n ) = T E ( n ) Φ( n −
1) = T E ( n, m ) Φ( m ) . To use this iteratively, one needs some initial condition. Particular initial conditions then leadto the Jost solutions u z ± . The suitable initial conditions are determined by the diagonalizationof the unperturbed transfer matrix T E = (cid:18) E − − (cid:19) , which is equal to T E ( n ) for n (cid:54)∈ {− K − + 1 , . . . , K + } . Recalling the relation (2), one has (cid:18) E − (cid:19) (cid:18) z z −
11 1 (cid:19) = (cid:18) z z −
11 1 (cid:19) (cid:18) z z − (cid:19) . This allows to read off the eigenvectors for the eigenvalues z and z − . Moreover, this motivatesto use the notations C z = (cid:18) z z −
11 1 (cid:19) , D z ( K ) = (cid:18) z K z − K (cid:19) . Now the above matrix identity can simply be written as T E C z = C z D z (1), which shows that( T E ) K C z = C z D z ( K ) . For the Jost solution u z + one should therefore choose Φ( K + ) = (cid:0) z (cid:1) up to a normalizationfactor, and for u z − rather Φ( K − ) = (cid:0) z (cid:1) . Adding suitable powers of z thus shows that theJost solutions are, for z ∈ C \ { } u z ± ( n ) = (cid:18) (cid:19) ∗ T E ( n, K ± ) (cid:18) z (cid:19) z K ± . As in (15), the Jost solutions allow construct matrices Φ z ± ( n ) ∈ C L × L spanning L -dimensionalplanes in C L : Φ z ± ( n ) = (cid:18) u z ± ( n + 1) u z ± ( n ) (cid:19) = T E ( n, K ± ) (cid:18) z (cid:19) z K ± . (16)8sing the above notations, one has (cid:0) Φ z ± ( n ) , Φ /z ± ( n ) (cid:1) = T E ( n, K ± ) C z D z ( K ± ) . (17)Note that for n ≥ K + one has (cid:0) Φ z + ( n ) , Φ /z + ( n ) (cid:1) = C z D z ( n ) = (cid:18) z n +1 z − n − z n z − n (cid:19) , (18)and similarly for (cid:0) Φ z − ( n ) , Φ /z − ( n ) (cid:1) when n ≤ K − .To conclude this section, let us briefly point out that it is also possible to construct thelinearly growing solutions v z ± for z = 1, for example: v ( n ) = (cid:18) (cid:19) ∗ T ( n, K + ) (cid:18) ( K + + 1) K + (cid:19) . Definition 11.
For two functions u, v : Z → C L × L and n ∈ Z , the Wronskian is defined by W n ( u, v ) = ı (cid:0) u ( n + 1) ∗ v ( n ) − u ( n ) ∗ v ( n + 1) (cid:1) ∈ C L × L . (19) If W n ( u, v ) is independent of n , it is simply denoted by W ( u, v ) . Clearly one has W n ( u, v ) ∗ = W n ( v, u ). The definition already suggests that the Wronskianis independent of n for functions u and v of interest. This basic fact follows from a shortcalculation: Lemma 12.
For a pair of matrix solutions Hu = Eu and Hv = Ev of the Schr¨odingerequation at complex conjugate energies, the Wronskian W n ( u, v ) is independent of n . Thus Wronskians like W ( u z − , u z + ) and W ( u /z − , u z + ) do not carry the index n . Moreover,they are analytic on C \ { } . The Wronskians of the Jost solutions can be evaluated explicitlyby using the constancy in n of (19), either for n > K + or n < K − . One finds the Wronskianidentities W ( u z ± , u z ± ) = 0 , W ( u /z ± , u z ± ) = ( ν z ) − . Using the notations (16) allows to rewrite the Wronskian as W ( u z + , u z ± ) = (Φ z + ) ∗ ı I Φ z ± , W ( u /z + , u z ± ) = (Φ /z + ) ∗ ı I Φ z ± . (20)The Wronskian identities then become(Φ /z ± , Φ z ± ) ∗ ı I (Φ z ± , Φ /z ± ) = ( ν z ) − J . (21)Here the index n in all Φ z ± ( n ) is dropped. In particular, these identities imply that thematrices (Φ z ± , Φ /z ± ) ∈ C L × L are invertible.The following Wronskian identity involving the derivatives of the Jost solutions w.r.t. z will also be used below. 9 emma 13. For z ∈ C \ { } , σ, η ∈ {− , + } and n ∈ Z , one has W n ( u zσ , ∂ z u zη ) = W n − ( u zσ , ∂ z u zη ) − ı (1 − z − ) u zσ ( n ) ∗ u zη ( n ) . A similar identity holds for W n ( u /zσ , ∂ z u zη ) . Proof.
The equation Hu z ± = ( z + z − ) u z ± at every point n ∈ Z reads u zσ ( n + 1) + V ( n ) u zσ ( n ) + u zσ ( n −
1) = ( z + z − ) u zσ ( n ) . (22)Taking the adjoint of this equation with z replaced by z and multiplying from the right by ∂ z u zη ( n ) leads to u zσ ( n +1) ∗ ∂ z u zη ( n ) + u zσ ( n ) ∗ V ( n ) ∂ z u zη ( n ) + u zσ ( n − ∗ ∂ z u zη ( n ) = ( z + z − ) u zσ ( n ) ∗ ∂ z u zη ( n ) . (23)Deriving equation (22) for σ replaced by η w.r.t. z gives ∂ z u zη ( n + 1) + V ( n ) ∂ z u zη ( n ) + ∂ z u zη ( n −
1) = (1 − z − ) u zη ( n ) + ( z + z − ) ∂ z u zη ( n ) . (24)Multiplying (24) from the left by u zσ ( n ) ∗ and then subtracting (23) leads to the claim. (cid:50) The plane wave transfer matrix M z is introduced in Definition 2. This terminology willbe explained and justified in this section. Recall the definition (3), namely ( u z − , u /z − ) =( u z + , u /z + ) M z . Using the matrices Φ z ± ( n ) ∈ C L × L in (16), this can be rewritten as (cid:0) Φ z − , Φ /z − (cid:1) = (cid:0) Φ z + , Φ /z + (cid:1) M z . Note that on both sides of this equality one can still replace the lattice site n . Multiplying theequation from the left with (Φ /z ± , Φ z ± ) ∗ ı I and taking into account the Wronskian identities(21) as well as J = shows M z = ν z J (cid:0) Φ /z + , Φ z + (cid:1) ∗ ı I (cid:0) Φ z − , Φ /z − (cid:1) . (25)It follows that M z is analytic in z away from {− , , } . In order to make connections withthe transfer matrix, let us now use (17) for the minus sign and n = K + followed by (18): T E ( K + , K − ) C z D z ( K − ) = (cid:0) Φ z − ( K + ) , Φ /z − ( K + ) (cid:1) = (cid:0) Φ z + ( K + ) , Φ /z + ( K + ) (cid:1) M z = C z D z ( K + ) M z , implying M z = (cid:0) C z D z ( K + ) (cid:1) − T E ( K + , K − ) (cid:0) C z D z ( K − ) (cid:1) . (26)10his suggests introducing the one-step plane wave transfer matrices at n ∈ Z by M z ( n ) = (cid:0) C z D z ( n ) (cid:1) − T E ( n ) (cid:0) C z D z ( n − (cid:1) , (27)as well as the several step version by M z ( n, m ) = M z ( n ) · · · M z ( m + 1) and M z ( m, n ) = M z ( n, m ) − for n > m , just as for the transfer matrices. Also let us set M z ( n, n ) = . Withthese notations, one has M z ( n, n −
1) = M z ( n ) and the transfer matrix of Definition 2 is M z = M z ( K + , K − ). One can now deduce a first crucial feature of M z ( n, m ), namely theirmultiplicativity still holds. Indeed, decomposing T E ( n, m ) = T E ( n ) T E ( n − , m ) leads to: M z ( n, m )= (cid:0) C z D z ( n ) (cid:1) − T E ( n ) (cid:0) C z D z ( n − (cid:1) (cid:0) C z D z ( n − (cid:1) − T E ( n − , m ) ( C z D z ( m ) (cid:1) = M z ( n ) M z ( n − , m )= M z ( n ) · · · M z ( m + 1) . (28)From (27), one can also calculate the plane wave transfer matrix explicitly: M z ( n ) = (cid:18) (cid:19) + ı ν z (cid:18) V ( n ) z − n V ( n ) − z n V ( n ) − V ( n ) (cid:19) . (29)Therefore M z ( n ) = if the potential V ( n ) vanished. From (29) one readily checks theidentities M /z ( n ) ∗ J M z ( n ) = J and M z ( n ) J M /z ( n ) ∗ = J . Combined with (28) onededuces the second set of crucial properties of the plane wave transfer matrices: M /z ( n, m ) ∗ J M z ( n, m ) = J , (30) M z ( n, m ) J M /z ( n, m ) ∗ = J . (31)For z ∈ S \{− , } , this shows that M z ( n, m ) is J -unitarity, namely it satisfies M ∗ J M = J .This also implies (7).In conclusion, the matrices M z ( n, m ) have the multiplicativity property and are J -unitarity, similar as the transfer matrices which are multiplicative and I -unitary. Further-more, they are trivial for vanishing potential and are thus adapted to plane waves. All thisjustifies the terminology used. The link to the transfer matrices T E ( n, m ) is established by(27). This shows that the passage from T E ( n, m ) to M z ( n, m ) is given by the basis changeinduced by C z , followed by the site-dependent diagonal factors (which destroy the represen-tation property). The basis change C z for z = ± ı induces via the M¨obius action the Cayleytransformation and this is well-known to induce a map from I -unitaries to J -unitaries ( e.g. [24]). This is also effect of the basis change here. For sake of concreteness, let us write out M z ( n, m ) in terms of the L × L block entries (depending on E ) of the transfer matrix: T E ( n, m ) = (cid:18) A BC D (cid:19) . (32)One finds M z ( n, m ) = ı ν z D z ( n ) − (cid:18) − B + C − zA + z − D − B + z − ( D − A ) + z − CB + z ( A − D ) − z C − C + B + z − A − zD (cid:19) D z ( m ) . z ∈ R , corresponding to energies E ∈ R \ ( − , M z ( n ) ∗ I M z ( n ) = I , (33)which is the I -unitary of M z ( n ) for real z ∈ R . This implies that also M z ( n, m ) and thus M z is I -unitary, which is item (ii) of Proposition 4. Let us begin by recalling the defining relation (4) of M z , namely (cid:0) u z − , u /z − (cid:1) = (cid:0) u z + , u /z + (cid:1) M z .Multiplying this by the inverse of M z leads to (cid:0) u z + , u /z + (cid:1) = (cid:0) u z − , u /z − (cid:1) ( M z ) − . Introducingnotations for the matrix coefficients( M z ) − = (cid:32) M z + N /z + N z + M /z + (cid:33) , the above two equations together with (4) imply u z + = u z − M z + + u /z − N z + , u /z − = u z + N z − + u /z + M z − , (34)holding for z ∈ C \ {− , , } . This can also be rewritten asΦ z + = Φ z − M z + + Φ /z − N z + , Φ /z − = Φ z + N z − + Φ /z + M z − . Therefore the Wronskian identities (21) lead to M z + = ν z (Φ /z − ) ∗ ı I Φ z + = ν z W ( u /z − , u z + ) ,N z + = − ν z (Φ z − ) ∗ ı I Φ z + = − ν z W ( u z − , u z + ) ,N z − = ν z (Φ /z + ) ∗ ı I Φ /z − = ν z W ( u /z + , u /z − ) ,M z − = − ν z (Φ z + ) ∗ ı I Φ /z − = − ν z W ( u z + , u /z − ) . (35)This shows that M z ± and N z ± are analytic on C \ {− , , } . Hence M z can also be writtenusing the Wronskians: M z = ν z (cid:32) W ( u /z + , u z − ) W ( u /z + , u /z − ) − W ( u z + , u z − ) − W ( u z + , u /z − ) (cid:33) . (36)Next let us note that ( M z ) − = J ( M /z ) ∗ J due to (31), which is equivalent to( N z + ) ∗ = − N /z − , ( M z + ) ∗ = M z − . (37)Furthermore, writing out the relations (30) and (31), a computation leads to: M z + ( M /z + ) ∗ = + N /z + ( N z + ) ∗ , ( M /z − ) ∗ M z − = + ( N /z − ) ∗ N z − , (38) M z + N z − = − N /z + M z − , M z − N z + = − N /z − M z + , (39)( M /z + ) ∗ M z + = + ( N /z + ) ∗ N z + , M z − ( M /z − ) ∗ = + N /z − ( N z − ) ∗ . (40)12ombined with (37) these equations imply for z ∈ R :( N z − ) ∗ M z − = ( M z − ) ∗ N z − , ( M /z − ) ∗ M z − = + ( N /z − ) ∗ N z − . (41) Lemma 14.
For z ∈ S \ {− , } , M z ± are invertible. Proof.
For z ∈ S , one has 1 /z = z and hence the identities (40) imply ( M z ± ) ∗ M z ± ≥ sothat M z ± are invertible. (cid:50) The scattering matrix was introduced in Definition 3. This section merely expresses thescattering matrix in terms of the matrix coefficients M z ± and N z ± , and then deduces some firstbasic properties. For that purpose, let z ∈ C so that the inverses ( M z ± ) − exist. Then onecan rewrite (34) as u z − = u z + ( M z + ) − − u /z − N z + ( M z + ) − , u /z + = u /z − ( M z − ) − − u z + N z − ( M z − ) − . Comparing with (5) in Definition 3, the scattering matrix at z ∈ C thus is S z = (cid:18) ( M z + ) − − N z − ( M z − ) − − N z + ( M z + ) − ( M z − ) − (cid:19) , (42)which allows to read off the transmission and reflection coefficients. Furthermore the equations(37) and (39) also allow to rewrite (42), e.g. S z = (cid:18) (( M z − ) ∗ ) − − N z − ( M z − ) − ( M z − ) − N /z − ( M z − ) − (cid:19) . (43)Based on these formulas (37) to (40), a calculation allows to verify the claims of Proposi-tion 4(ii). Furthermore, Proposition 4(v) also follows because z (cid:55)→ M z − is meromorphic due to(35)) and therefore z (cid:55)→ det( M z − ) also. Moreover, M z ± → for z →
0, and the limits ( M z ± ) − exist when z → ± M z − ) − form a discrete set. Hence also z (cid:55)→ ( M z − ) − is meromorphicand so is z (cid:55)→ S z . In this section, E ∈ R \ [ − , z ∈ R \ {− , , } . If | z | <
1, then the Jost solution u z + is decaying exponentially at + ∞ , and the Jost solution u /z − decays exponentially at −∞ .If these two solutions match together, one gets eigenstates for the selfadjoint operator H . Ofcourse, such eigenstates cannot appear for complex energies. Proposition 15.
For E = z + z − (cid:54)∈ σ ( H ) with | z | < , M z ± are invertible. roof. Recall from (3) that u /z − = u z + N z − + u /z + M z − . Let us assume that M z ± is not invertibleand then show that E ∈ σ ( H ). If M z − has a non-trivial kernel, then the left and right solutionsmatch on a subspace which produces square integrable eigenstates, so that indeed E ∈ σ ( H ).Now the invertibility of M z + follows from (37). (cid:50) Let us now look more closely at the multiplicity of the eigenvalues. The intersection of thespace of left decreasing and right decreasing solution lead to square integrable bound state atenergy E . More precisely, this intersection can be parametrized byRan (cid:0) Φ z + ( K + ) (cid:1) ∩ Ran (cid:0) T E ( K + , K − )Φ /z − ( K − ) (cid:1) . (44)Provided the intersection is non-trivial, one can then choose φ ∈ C L such that Φ z + ( K + ) φ is inthe intersection, and this provides the bound state u φ ( n ) = (cid:18) (cid:19) ∗ T E ( n, K + )Φ z + ( K + ) φ . Remark 16.
Replacing (37) into (38), one finds( M /z − ) ∗ M z − = − N z + N z − , , ( M /z + ) ∗ M z + = − N z − N z + . This implies that N z + N z − φ = φ for φ ∈ Ker( M z − ), and N z − N z + φ = φ for φ ∈ Ker( M z + ). More-over, equations (39) imply that N z − (Ker( M z − )) ⊂ Ker( M z + ) and N z + (Ker( M z + )) ⊂ Ker( M z − ).Therefore N z − (cid:12)(cid:12) Ker ( M z − ) : Ker( M z − ) → Ker( M z + ) (45)is an isomorphism with inverse N z + | Ker ( M z + ) . (cid:5) Proposition 17.
For E = z + z − ∈ R with | z | < , one has multiplicity of E as eigenvalue of H = dim Ker( M z ± )= order of z as zero of z (cid:48) (cid:55)→ det( M z (cid:48) ± ) . In particular, M z ± is invertible if z + z − is not an eigenvalue. Proof.
The argument in the proof of Proposition 15 and the discussion above imply the firstequality. For the proof of the second equality let us notice that z + 1 /z ∈ R with | z | < z ∈ R . Let p denote the dimension of Ker( M z − ) and let { w , . . . , w p } be a basis of Ker( M z − ). Since N z − : Ker( M z − ) → Ker( M z + ) is an isomorphism by Remark16, it follows that { N z − w , . . . , N z − w p } is a basis of Ker( M z + ) = Ker(( M z − ) ∗ ) = Ran( M z − ) ⊥ (see (37)). Let { v p +1 , . . . , v L } be an orthonormal basis of Ran( M z − ) and { u p +1 , . . . , u L } vectors such that M r − u j = v j for j ∈ { p + 1 , . . . , L } . Then { w , . . . , w p , u p +1 , . . . , u L } and { N z − w , . . . , N z − w p , v p +1 , . . . , v L } are bases of C L . Let us introduce the invertible L × L ma-trices U = (cid:0) w , . . . , w p , u p +1 , . . . , u L (cid:1) , V = (cid:0) N z − w , . . . , N z − w p , v p +1 , . . . , v L (cid:1) .
14t follows that V ∗ M z − U = (cid:18) (cid:19) . Since ζ (cid:55)→ M ζ − is analytic around z , one has M ζ − = M z − + ( ζ − z ) ∂ z M z − + O (( ζ − z ) ) so that V ∗ M ζ − U = (cid:18) ( ζ − z ) V ∗ ∂ z M z − U ( ζ − z ) B ( ζ − z ) C + ( ζ − z ) D (cid:19) + O (cid:0) ( ζ − z ) (cid:1) , for some constant matrices B, C, D and where U , V are given by U = (cid:0) w , . . . , w p (cid:1) , V = (cid:0) N z − w , . . . , N z − w p (cid:1) = N z − U . The matrix V ∗ ∂ z M z − U is invertible because Lemma 18 implies that for z > φ ∈ C p φ ∗ V ∗ ∂ z M z − U φ = ( V φ ) ∗ ∂ z M z − U φ = ( U φ ) ∗ ( N z − ) ∗ ∂ z M z − ( U φ ) > , and similarly for z <
0. Using the Schur complement formula for the determinant, one obtainsdet( V ∗ M ζ − U ) = det (cid:0) + ( ζ − z ) D + O (( ζ − z ) ) (cid:1) det (cid:0) ( ζ − z )( V ∗ ∂ z M z − U + O ( ζ − z )) (cid:1) = ( ζ − z ) p g ( ζ ) , with g being a function satisfying g ( z ) = det( V ∗ ∂ z M z − U ) (cid:54) = 0. This implies the claim. (cid:50) Lemma 18.
For z ∈ ( − , and φ ∈ Ker( M z − ) , φ ∗ ( N z − ) ∗ ∂ z M z − φ = z − (cid:107) u /z − φ (cid:107) . Proof:
First of all note that for φ ∈ Ker( M z − ), (34) implies that u /z − φ = u z + N z − φ which ishence a square summable vector both at −∞ and ∞ . Consequently the (cid:96) -norm (cid:107) u /z − φ (cid:107) appearing in the statement is indeed finite.Let us start from M z − = − ν z W ( u z + , u /z − ) given in (35) with z = z ∈ R . Deriving leads to ∂ z M z − = − ( ∂ z ν z ) W ( u z + , u /z − ) − ν z W n ( ∂ z u z + , u /z − ) − ν z W n ( u z + , ∂ z u /z − ) , where n ∈ Z is arbitrary. As W ( u z + , u /z − ) φ = 0 and u /z − φ = u z + N z − φ , this implies φ ∗ ( N z − ) ∗ ∂ z M z − φ = − ν z φ ∗ ( N z − ) ∗ W n ( ∂ z u z + , u /z − ) φ − ν z φ ∗ ( N z − ) ∗ W n ( u z + , ∂ z u /z − ) φ = − ν z φ ∗ ( N z − ) ∗ W n ( ∂ z u z + , u z + ) N z − φ − ν z φ ∗ W n ( u /z − , ∂ z u /z − ) φ . As | z | <
1, one has lim n →∞ W n ( ∂ z u z + , u z + ) = 0 . To compute the limit of the other contribution let us invoke Lemma 13 iteratively. As ∂ z f (1 /z ) = − z − ∂ /z f (1 /z ) for every analytic function, one computes φ ∗ ( N z − ) ∗ ∂ z M z − φ = z − ν z lim n →∞ (cid:32) φ ∗ W k ( u /z − , ∂ /z u /z − ) φ − n (cid:88) m = k +1 ı (1 − z ) φ ∗ u /z − ( m ) ∗ u /z − ( m ) φ (cid:33) . This holds for any k . As lim k →−∞ W k ( u /z − , ∂ /z u /z − ) = 0 , the claim now follows in the limit k → −∞ . (cid:50) orollary 19. Recall that C is the set of points z ∈ C \ {− , , } where M z ± are invertible.The set C contains an open neighborhood of the unit disc with the points {− , , } and { z ∈ C : z + z − ∈ σ p ( H ) } removed. Proof.
For z ∈ S \ {− , } the invertibility of M z − is stated in Lemma 14. For (cid:61) m ( z ) (cid:54) = 0this is Proposition 15. For real z (cid:54) = 0 the invertibility is characterized in Proposition 17. (cid:50) Corollary 19 contains no statement about the analyticity of M z − in {− , , } . Section 8will consider z = 0 and Sections 9 and 10 the points z = ±
1. The remainder of this sectionconsists of comments about geometric structures behind Proposition 17. They freely use thenotion of intersection theory of Lagrangian planes and some of the terminology linked to thetheory of the Maslov index, see e.g. [24, 25]. Let us stress that these remarks are not relevantfor the following though.
Remark 20.
This remark provides alternative proofs of the two identities in Proposition 17.The intersection (44) is actually the intersection of two I -Lagrangian planes and hence itsdimension J z can be calculated by intersection theory ( e.g. Proposition 2 in [25]) as J z = dim Ker (cid:16) Φ z + ( K + ) ∗ I T E ( K + , K − )Φ /z − ( K − ) (cid:17) = dim Ker (cid:16) ν z Φ z + ( K + ) ∗ ı I Φ /z − ( K + ) (cid:17) = dim Ker (cid:18)(cid:18) (cid:19) ∗ M z (cid:18) (cid:19)(cid:19) = dim Ker (cid:0) M z − (cid:1) , where (25) was used. This shows the first identity. For the second equality, let us recall from(37) that ( M z + ) ∗ = M z − . Hence the analytic matrix function H z = (cid:18) M z − M z + (cid:19) is selfadjoint for z ∈ ( − , − , z ∈ ( − , (cid:55)→ det( H z ) = det( M z − ) det( M z + ) = | det( M z − ) | has a zeroof order at least 2 dim(Ker( M z − )) = 2 dim(Ker( M z + )). To verify that the zero is of this or-der 2 dim(Ker( M z ± )), it will be shown that the restriction of the selfadjoint matrix ∂ z H z toKer( H z ) = Ker( M z + ) ⊕ Ker( M z − ) is non-degenerate. This follows from the Max-Min principleby showing that Ker( H z ) has two subspaces of dimension dim(Ker( M z ± )), on one of which ∂ z H z is positive definite and on the other of which it is negative definite.For this purpose, let us recall from the remark above that N z − : Ker( M z − ) → Ker(( M z − ) ∗ ) =Ker( M z + ) is an isomorphism. Hence for φ ∈ Ker( M z − ), one has N z − φ ∈ Ker( M z + ) and therefore N z − φ ⊕ σφ ∈ Ker( H z ) for σ ∈ {− , } . Now (cid:18) N z − φσφ (cid:19) ∗ ∂ z H z (cid:18) N z − φσφ (cid:19) = σ φ ∗ ( N z − ) ∗ ∂ z M z − φ + σ φ ∗ ∂ z M z + N z − φ = 2 σ φ ∗ ( N z − ) ∗ ∂ z M z − φ , where the second equality follows from (37) and the selfadjointness of ( N z − ) ∗ M z − , see (41).Therefore the proof of the the second equality is again completed by Lemma 18. (cid:5) emark 21. For z ∈ ( − , ∪ (0 , (cid:0) N z − M z − (cid:1) span an I -Lagrangian plane in C L , namely an L -dimensional subspaceon which I viewed as sesquilinear quadratic form vanishes. One can then introduce the phaseof this plane via the stereographic projection: U z = ( N z − − ı M z − )( N z − + ı M z − ) − . The inverse of N z − + ı M z − indeed exists and U z is unitary. Now according to Proposition 17,the multiplicity of E = z + z − as eigenvalue of H is given by the dimension of the intersectionof the two I -Lagrangian planes (cid:0) N z − M z − (cid:1) and (cid:0) (cid:1) . This dimension is equal the dimension of z = 1as eigenvalue of U z because ψ = ( N z − + ı M z − ) − φ ∈ Ker( U z − ) ⇐⇒ φ ∈ Ker( M z − ) . Furthermore, the identity ı ( U z ) ∗ ∂ z U z = − (cid:0) ( N z − + ı M z − ) − (cid:1) ∗ (cid:0) ( N z − ) ∗ ∂ z M z − − ( M z − ) ∗ ∂ z N z − (cid:1) ( N z − + ı M z − ) − combined with Lemma 18 shows that for all ψ = ( N z − + ı M z − ) − φ ∈ Ker( U z − ) ψ ∗ ı ( U z ) ∗ ∂ z U z ψ = 2 z − (cid:107) u /z − φ (cid:107) . In particular, ı ( U z ) ∗ ∂ z U z restricted to the kernel of U z − is definite with a sign given bythe sign of z . This implies that the path z (cid:55)→ (cid:0) N z − M z − (cid:1) is transversal to (cid:0) (cid:1) and unidirectional.Therefore the associated Maslov index is the spectral flow of z (cid:55)→ U z through 1 and countsexactly the eigenvalues of H . Let us also note that due to Proposition 22 proved below thereis a neighborhood of z = 0 with no intersections. (cid:5) Let us note that z → | E | → ∞ . Hence the following statement concerns thehigh energy asymptotics. Proposition 22.
One has lim z → M z − = . Thus z = 0 is a removable singularity of M z − . Proof:
Due to the factorization property (28), M z = K + (cid:89) n = K − M z ( n ) , where here and in the following the product is ordered with the factors ordered according tothe index, smallest n being on the r.h.s.. Each matrix M z ( n ) is given by (29) and will befactorized as follows: M z ( n ) = + ı ν z (cid:18) V ( n ) 00 V ( n ) (cid:19) (cid:20) J + (cid:18) z − n − z n (cid:19) (cid:21) . M z = + (cid:88) ∅(cid:54) = J ⊂ [ K − +1 ,K + ] ( ıν z ) | J | (cid:34)(cid:89) n ∈ J (cid:18) V ( n ) 00 V ( n ) (cid:19)(cid:35) (cid:89) n ∈ J (cid:20) J + (cid:18) z − n − z n (cid:19) (cid:21) , where the sum is over all subsets J of { K − + 1 , . . . , K + } and | J | denotes the cardinality of J . Now by (4) M z − is the lower right entry of M z . Multiplying out the products on ther.h.s. leads to a large number of summands, but in each there will be ordered factors of theoff-diagonal matrix of the type (cid:18) z − n − z n (cid:19) (cid:18) z − m − z m (cid:19) = (cid:18) − z m − n ) − z n − m ) (cid:19) , n > m , or possibly with a factor J between the first two factors. The crucial fact is now that thelower right entry is non-singular, as due to the ordering one has n > m . Therefore all lowerright entries are non-singular. Moreover, lim z → ν z = 0. This implies the claim. (cid:50) At the band edges E = ±
2, one has z = ±
1. As z → ± ν z → ∞ with apole of first order. Due to (29) and (28), this leads to singularities of the plane wave transfermatrix M z . On first sight, one may expect these singularities to be of order K + − K − ,stemming from the multiplication of the factors ν z . However, the special structure of (29)implies that the singularity is only of order 1. Proposition 23.
There exists an anlatytic function z ∈ C \ { } (cid:55)→ G z ± ∈ C L × L and matrices F ± ∈ C L × L such that M z = G z ± + ν z F ± in a neighborhood of ± . Proof.
Let us provide two proofs. First consider the set G = (cid:26)(cid:18) V z n V − z m V − z n +2 m V (cid:19) : V ∈ C L × L , n, m ∈ Z (cid:27) . Note that V is not necessarily selfadjoint here. The set G is a multiplicative subsemigroup of C L × L because for any V, V (cid:48) ∈ C L × L and n, m, n (cid:48) , m (cid:48) ∈ Z , one has the identity (cid:18) V z n V − z m V − z n +2 m V (cid:19) (cid:18) V (cid:48) z n (cid:48) V (cid:48) − z m (cid:48) V (cid:48) − z n (cid:48) +2 m (cid:48) V (cid:48) (cid:19) = (1 − z n +2 m (cid:48) ) (cid:18) V V (cid:48) z n (cid:48) V V (cid:48) − z m V V (cid:48) − z n (cid:48) +2 m V V (cid:48) (cid:19) . By (29) and (28), M z is in the span of G . Moreover, for n + m (cid:48) >
0, each such productcontains a factor (1 − z n +2 m (cid:48) ) = (1 − z )(1 + z + . . . + z n +2 m (cid:48) − ). As ν z (1 − z ) = − ız , thiscancels the singularity of one ν z . For n + m (cid:48) <
0, one argues similarly and for n + m (cid:48) = 0 theproduct vanishes. Consequently, in all products in (28) only one singular factor ν z remains.Extracting its singularity leads to the claim. 18he second proof is based on (26) and an explicit calculation of the inverse of C z whichshows M z = ı ν z D z ( K + ) − (cid:18) − z − − z (cid:19) T E ( K + , K − ) C z D z ( K − ) . Therefore one can compute the limit F ± = lim z →± ( ν z ) − M z = ı (cid:18) − ± ∓ (cid:19) T ( K + , K − ) (cid:18) ± ±
11 1 (cid:19) , and deduce G z ± = M z − ν z F ± . (cid:50) The matrices F ± in Proposition 23 can also be expressed in terms of Wronskians of Jostsolutions. Indeed, comparing with (36), one finds F ± = lim z →± ( ν z ) − M z = lim z →± (cid:32) W ( u /z + , u z − ) W ( u /z + , u /z − ) − W ( u z + , u z − ) − W ( u z + , u /z − ) (cid:33) . Furthermore Proposition 23 implies that M z − as a matrix entry of M z has a similar singularitybehavior, namely in a neighborhood of ± M z − = G z ± + ν z F ± , for trigonometric polynomials G z ± ∈ C L × L and matrices F ± ∈ C L × L . Please note again thatthe subscripts ± correspond to the upper/lower band edge at ± M z − . Comparing with the above, one deduces F ± = lim z →± − W ( u z + , u /z − ) = − W ( u , u − ) . (46)For further analysis, let us focus on F = F + at the upper band edge. The lower band edgeis similar. The singular value decomposition is F = U DU (cid:48) where U and U (cid:48) are unitary and D ≥ F is not singular so that D >
0, one has T − = lim z → (cid:0) G z + ν z F (cid:1) − = lim z → ( U (cid:48) ) ∗ (cid:0) U ∗ G z ( U (cid:48) ) ∗ + ν z D (cid:1) − U ∗ = 0 . Also the coefficient matrix N z − has a decomposition N z − = ˆ G z + ν z ˆ F withˆ F = lim z → W ( u /z + , u /z − ) = W ( u , u − ) . Consequently one finds in the non-singular case R − = − lim z → N z − ( M z − ) − = − lim z → ( ˆ G z + ν z ˆ F )( G z + ν z F ) − = − ˆ F F − = . (47)Next let us turn to the case of a singular matrix F . Then D = diag(0 , f ) with f >
0. Forlater use, let us set J h = L − rank( f ) = L − rank( F ) = dim Ker( F ). Then T z − = ( G z + ν z U DU (cid:48) ) − = ( U (cid:48) ) ∗ ( U ∗ G z ( U (cid:48) ) ∗ + ν z diag(0 , f )) − U ∗ . D ).Set U ∗ G z ( U (cid:48) ) ∗ = (cid:18) a z b z c z d z (cid:19) . Then U (cid:48) T z − U = (cid:18) ( s z ) − − ( s z ) − b z ( d z + ν z f ) − − ( d z + ν z f ) − c z ( s z ) − ( d z + ν z f ) − + ( d z + ν z f ) − c z ( s z ) − b z ( d z + ν z f ) − (cid:19) , (48)with Schur complement s z = a z − b z ( d z + ν z f ) − c z . Recall the general fact the Schur comple-ment of a matrix with invertible lower right entry is invertible if and only if the matrix itselfis invertible. Here the matrix U (cid:48) T z − U = U (cid:48) ( M z − ) − U is invertible for z ∈ S = S \ {− , } .Moreover, (cid:107) U (cid:48) T z − U (cid:107) = (cid:107) T z − (cid:107) = (cid:107) ( M z − ) − (cid:107) ≤ z ∈ S due to ( M z − ) ∗ M z − ≥ (see the proofof Lemma 14). Therefore also (cid:107) ( s z ) − (cid:107) ≤ z ∈ S . On the other hand, f > z → s z = a so that a is invertible. It follows T − = ( U (cid:48) ) ∗ (cid:18) ( a ) −
00 0 (cid:19) U ∗ . Furthermore, one has R z − = ( ˆ G z + ν z ˆ F ) T z − . As the limit T − is bounded and (cid:107) R z − (cid:107) ≤
1, onemust have ˆ
F T − = 0 so that R − = ˆ G T − . Formulas for T and R can be obtained in a similar manner. Another set of formulas forthe limits result from the identities (35), notably T = lim z → ı ( z − − z ) W ( u /z − , u z + ) − , T − = lim z → ı ( z − z − ) W ( u z + , u /z − ) − , and similarly for R ± . This implies T − = lim z → ı ( z − z − ) W ( u z + , u /z − ) − = (cid:16) lim z → ı ( z − − z ) W ( u /z − , u z + ) − (cid:17) ∗ = ( T ) ∗ . Similarly, one checks R − = ( R ) ∗ . Analogous formulas for z = − M z − ) − = T z − , this implies the following fact that will be used later on. Proposition 24.
The limits lim z →± ( M z − ) − exist.
10 Half-bound states
At the band edge E = 2 (and similarly E = − z = 1. In the limit z →
1, thesolutions u ¯ z + and u /z − can possibly aline and then their Wronskian W ( u ¯ z + , u /z − ) converges to 0,at least on a subspace. Hence its inverse will diverge there. In the last section, it was shownthat the associated pole can be extracted and determines the limit of the scattering matrix.The non-generic behavior where (12) does not hold, is connected to the existence of special20tates at band edge energies. Indeed, if the Wronskian W ( u ¯ z + , u /z − ) has a kernel in the limit z →
1, then the alignment of directions of u ¯ z + and u /z − in the limit z → (cid:96) ∞ ( Z , C L ) and thus the span of the bounded (Jost) solutions u ± at ±∞ . Similar as in (16) they lead to matricesΦ ± ( n ) = (cid:18) (cid:19) ∈ C L × L , n > K + or n < K − respectively . These matrices satisfy (Φ ± ) ∗ I Φ ± = 0 and are thus I -Lagrangian. Asymptotically constantsolutions (both at + ∞ and −∞ ) are constructed from vectors inRan (cid:0) Φ ( K + ) (cid:1) ∩ Ran (cid:0) T ( K + , K − )Φ − ( K − ) (cid:1) , (49)provided the intersection is non-trivial. Then choosing φ ∈ C L such that Φ ( K + ) φ is in theintersection, one then has a bounded solution u φ ( n ) = (cid:18) (cid:19) ∗ T ( n, K + )Φ ( K + ) φ . Such solutions are constant outside of the support { K − + 1 , . . . , K + } . In the case of contin-uous Schr¨odinger operators, such solutions are called half-bound states because they have acontribution in Levinson’s theorem. These states are not in the Hilbert space, but may,depending on dimension, still decay [12]. Here theses solutions are asymptotically constant.Of course, these half-bound states do not lead to eigenvalues of H as a selfadjoint operator.Now let J + h be the dimension of this intersection. By construction, it is equal to thedimension of the space of bounded solutions at energy E = 2 and thus z = 1 (that is, thedimension of half-bound states at z = 1). There is a similar dimension J − h for E = − z = − Proposition 25.
One has J ± h = dim Ker( F ± ) with F ± as defined in Section 9 at z = ± .Moreover, the map z (cid:55)→ det( M z − ) has a pole of order L − J ± h at z = ± . Proof.
Let us focus on z = 1. By (46), F + = − W ( u , u − ) = ı ( u − ( n + 1) − u − ( n )) , for all n ≥ K + . It follows thatlim n →∞ u − ( n ) n = lim n →∞ (cid:80) n − m =0 ( u − ( m + 1) − u − ( m )) n = − ı F + . As ( u , v ) is a fundamental solution at E = 1, there are L × L matrices X, Y such that u − = u X + v Y . Using that u ( n ) = and v ( n ) = n for n ≥ K + , it follows that Y = − ıF + .Therefore, u − ( n ) = X − ınF + for n ≥ K + . Therefore solutions of the form u − φ are boundedif and only if φ ∈ Ker( F + ). This implies that J + h = dim(Ker( F + )).As to the second claim, it is equivalent to z (cid:55)→ det(( M z − ) − ) = det( T z − ) having a zero oforder L − J ± h at z = ±
1. This will be verified for z = 1 using the explicit formulas for T z − z = − U, U (cid:48) as given there, let us introducethe following notation for the matrix entries of U (cid:48) T z − U as given in (48): U (cid:48) T z − U = (cid:18) ˆ a z ˆ b z ˆ c z ˆ d z (cid:19) , namely ˆ a z = ( s z ) − , ˆ b z = − ( s z ) − b z ( d z + ν z f ) − and so on. As f > (cid:107) d z (cid:107) ≤ z ∈ S ,one has lim z → ( d z + ν z f ) − = 0 , lim z → ν z ( d z + ν z f ) − = f − . This leads tolim z → ˆ b z ( ˆ d z ) − ˆ c z = lim z → ( s z ) − b z ( d z + ν z f ) − (cid:0) + c z ( s z ) − b z ( d z + ν z f ) − (cid:1) − c z ( s z ) − = 0 . Notice that ˆ d z is invertible because ν z ˆ d z → f − by (48). Hencelim z → det (cid:0) ˆ a z − ˆ b z ( ˆ d z ) − ˆ c z (cid:1) = lim z → det (cid:0) ( s z ) − (cid:1) = det (cid:0) ( a ) − (cid:1) (cid:54) = 0 , because a is invertible. Therefore ˆ a z − ˆ b z ( ˆ d z ) − ˆ c z is invertible in a neighborhood of 1. Simi-larly, lim z → det (cid:0) ν z ˆ d z (cid:1) = det( f − ) . Finally, using Schur formula for the determinant,det( U (cid:48) T z − U ) = ( ν z ) − L + J + h det (cid:0) ν z ˆ d z (cid:1) det (cid:0) ˆ a z − ˆ b z ( ˆ d z ) − ˆ c z (cid:1) , because the size and rank of f is L − J + h . Due to the asymptotics stated above and the factthat ν z has a pole of order 1, this implies the result. (cid:50) Remark 26.
Let us provide another proof of J ± h = dim Ker( F ± ). It is of geometric natureand similar to the one in Remark 20. The intersection (49) is again the intersection oftwo I -Lagrangian planes so that its dimension J + h can be calculated by intersection theory(Proposition 2 in [25]) as J + h = dim Ker (cid:0) Φ ( K + ) ∗ I T ( K + , K − )Φ − ( K − ) (cid:1) . But as Φ ( K + ) ∗ = lim z → ( ı ν z ) − (cid:18) (cid:19) ∗ ( C z ) − I ∗ , Φ − ( K − ) = lim z → C z (cid:18) (cid:19) , one has due to (26)Φ ( K + ) ∗ I T ( K + , K − )Φ − ( K − ) = lim z → ( ıν z ) − (cid:18) (cid:19) ∗ M z (cid:18) (cid:19) = lim z → ( ıν z ) − M z − = − ı F , where F = F + is as in Section 9, which then also implies J + h = dim Ker( F ). (cid:5) Proposition 27.
For each φ ∈ Ran( T ) , there exists a state u φ ∈ (cid:96) ∞ ( Z , C L ) satisfying theSchr¨odinger equation Hu φ = 2 u φ . The limits lim n →±∞ u φ ( n ) exist. The dimension J + h of the space of these half-bound states is equal to dim(Ran( T )) . Asimilar statement hold for z = − .
11 Time delay
Levinson’s theorem concerns the winding of the scattering matrix on the unit circle. Theintegrand is also called the time delay. It can then be calculated by the following generalprinciple related merely to the passage (10) from the J -unitary M to the unitary V ( M ). Proposition 28.
Let t (cid:55)→ M t be a differentiable path of J -unitaries with diagonal entries A t and D t . Then Tr (cid:0) V ( M t ) ∗ ∂ t V ( M t ) (cid:1) = Tr (cid:0) ( A t ) − ∂ t A t − ( D t ) − ∂ t D t (cid:1) . Proof.
Let us drop the index t and also simply write ∂ = ∂ t . The J -unitarity of M and M ∗ is equivalent to the following identities implies A ∗ A = + C ∗ C , D ∗ D = + B ∗ B , A ∗ B = C ∗ D ,AA ∗ = + BB ∗ , DD ∗ = + CC ∗ , AC ∗ = BD ∗ . As already noted, A and D are thus invertible. NowTr (cid:0) V ( M ) ∗ ∂ V ( M ) (cid:1) = Tr (cid:18)(cid:18) A − A − B − ( D ∗ ) − B ∗ ( D ∗ ) − (cid:19) ∂ (cid:18) ( A ∗ ) − − BD − B ∗ ( A ∗ ) − D − (cid:19)(cid:19) = Tr (cid:0) A − ∂ ( A ∗ ) − + A − B∂B ∗ ( A ∗ ) − + A − BB ∗ ∂ ( A ∗ ) − + ( D ∗ ) − B ∗ ∂BD − + ( D ∗ ) − B ∗ B∂D − + ( D ∗ ) − ∂D − (cid:1) . Now let us replace BB ∗ and B ∗ B by the above expressions in the third and fifth summands:Tr (cid:0) V ( M ) ∗ ∂ V ( M ) (cid:1) = Tr (cid:0) A ∗ ∂ ( A ∗ ) − + A − B∂B ∗ ( A ∗ ) − + ( D ∗ ) − B ∗ ∂BD − + D∂D − (cid:1) = Tr (cid:0) A ∗ ∂ ( A ∗ ) − + ( A ∗ ) − A − B∂B ∗ + D − ( D ∗ ) − B ∗ ∂B + D∂D − (cid:1) = Tr (cid:0) A ∗ ∂ ( A ∗ ) − + ( AA ∗ ) − B∂B ∗ + ( D ∗ D ) − B ∗ ∂B + D∂D − (cid:1) . Now replace AA ∗ and D ∗ D in terms of B and use ( + B ∗ B ) − B ∗ = B ∗ ( + BB ∗ ) − . Againusing the cyclicity one findsTr (cid:0) V ( M ) ∗ ∂ V ( M ) (cid:1) = Tr (cid:0) A ∗ ∂ ( A ∗ ) − + ( + BB ∗ ) − ∂ ( BB ∗ ) + D∂D − (cid:1) = Tr (cid:0) A ∗ ∂ ( A ∗ ) − + ( AA ∗ ) − ∂ ( AA ∗ ) + D∂D − (cid:1) = Tr (cid:0) − ( A ∗ ) − ∂A ∗ + ( AA ∗ ) − ( ∂AA ∗ + A∂A ∗ ) − D − ∂D (cid:1) , (cid:50) When applied to (11), one gets a formula for the time delay for z ∈ S \ {− , } . As bothsides are meromorophic, this formula extends to all { z ∈ C : z, z − ∈ C } . Corollary 29.
For all z, z − ∈ C ∪ {− , } , Tr (cid:0) ( S /z ) ∗ ∂ z S z (cid:1) = Tr (cid:0) ( M /z − ) − ∂ z M /z − − ( M z − ) − ∂ z M z − (cid:1) (50)= det( M /z − ) − ∂ z det( M /z − ) − det( M z − ) − ∂ z det( M z − ) . (51)
12 Levinson-type theorem
This section is only devoted to the
Proof of Theorem 9.
Let us first prove the statements about the spectrum of H . Due toProposition 17 the eigenvalues are the zeros of z (cid:55)→ det( M z − ) and by Proposition 24 therecannot be any accumulation of eigenvalues at − H is bounded, thereare indeed only a finite number of eigenvalues outside of [ − , − ,
2] will have have oscillating eigenfunctions outside of the support of V so that they actually have to vanish outside of this support. However, compactly supportedeigenfuntions vanish identically due to the three-term recurrence equation resulting from (1).Let now S ± (cid:15) be the the positively oriented circle of radius 1 ± (cid:15) . Let us integrate (50) overthis path. Then (cid:73) S − (cid:15) dz πı Tr (cid:0) ( S /z ) ∗ ∂ z S z (cid:1) = − (cid:32)(cid:73) S (cid:15) + (cid:73) S − (cid:15) (cid:33) dz πı det( M z − ) − ∂ z det( M z − ) . Now one can invoke the argument principle. According to Proposition 17, Corollary 19 andProposition 25, the map z (cid:55)→ det( M z − ) is meromorphic on a neighborhood of the unit discwith zeros at z such that z + z − is an eigenvalue both counted with their multiplicity andwith poles of order L − J ± h at z = ±
1, and no other zeros or poles. Therefore the integralscan be evaluated: (cid:73) S − (cid:15) dz πı Tr (cid:0) ( S /z ) ∗ ∂ z S z (cid:1) = L − J + h − J b + L − J − h − J b , for (cid:15) sufficiently small. Now the integrand on the l.h.s. is analytic in a neighborhood of the S (see Theorem 8) so that one can take the limit (cid:15) →
0. As on 1 /z = z on S , (cid:73) S dz πı Tr (cid:0) ( S z ) ∗ ∂ z S z (cid:1) = 2 L − J h − J b , where J h = J + h + J − h is the total number of half-bound states. Finally let us split S intoupper and lower arcs S and S − , both with positive orientation. Parametrizing the upper arcwith k ∈ [0 , π ] (cid:55)→ e ık ∈ S followed by the change variables k ∈ [0 , π ] (cid:55)→ E = 2 cos( k ) shows (cid:73) S dz πı Tr (cid:0) ( S z ) ∗ ∂ z S z (cid:1) = (cid:90) π dk πı Tr (cid:0) ( S e ık ) ∗ ∂ k S e ık (cid:1) = (cid:90) − dE πı Tr (cid:0) ( S E ) ∗ ∂ E S E (cid:1) ,
24s according to our convention S E = S e ık . As to the lower circle, let us use the parametrization[ − π, (cid:55)→ e ık and the identities KS e ık K = ( S e − ık ) ∗ and K ∂ k S e ık K = ( ∂ k S e − ık ) ∗ , so that dueto the cyclicity of the trace and K = : (cid:73) S − dz πı Tr (cid:0) ( S z ) ∗ ∂ z S z (cid:1) = (cid:90) − π dk πı Tr (cid:0) ( S e ık ) ∗ ∂ k S e ık (cid:1) = (cid:90) − π dk πı Tr (cid:0) S e − ık ∂ k ( S e − ık ) ∗ (cid:1) = − (cid:90) − π dk πı Tr(( S e − ık ) ∗ ∂ k S e − ik )= (cid:90) π dk πı Tr(( S e ık ) ∗ ∂ k S e ik )= (cid:90) S dz πı Tr(( S z ) ∗ ∂ z S z ) , where in the third equality results from Tr(( S e − ık ) ∗ ∂ k S e − ık ) = − Tr( ∂ k ( S e − ık ) ∗ S e − ık ), which inturn follows from Proposition 4(iii) for 1 /z − = z ∈ S . Therefore the contribution of S − isthe same as that of S and this concludes the proof. (cid:50)
13 Green function and scattering matrix
Recall the definition (13) of the Green function. Note that (cid:61) m ( G E ( n, n )) > (cid:61) m ( E ) > z (cid:55)→ G z + z − ( n, m ) is analytic on C \ S .The following can readily be deduced from [23], but for sake of completeness we provide acomplete proof. Proposition 30.
For E = z + z − (cid:54)∈ σ ( H ) with | z | < , M z − = z K + − K − z − z − G E ( K − , K + ) − ,N z − = z − K + − K − G E ( K + , K + ) G E ( K − , K + ) − − z − K + − K − z − z − G E ( K − , K + ) − . Proof.
Let us set G E ( n ) = G E ( n, K + ) and view it as a matrix-valued function on Z . Then G E ( n + 1) + G E ( n −
1) + ( V ( n ) − E ) G E ( n ) = δ n,K + L . Let us introduce Ψ E ( n ) = (cid:18) G E ( n + 1) G E ( n ) (cid:19) . Using the three term recurrence relation iteratively, one sees that Ψ E ( n ) is of full rank L .Furthermore, one has Ψ E ( n ) = T E ( n )Ψ E ( n −
1) + δ n,K + (cid:18) (cid:19) .
25n particular, Ψ E ( K + ) = T E ( K + , K − ) Ψ E ( K − ) + (cid:18) (cid:19) . (52)As the matrix Ψ E ( n ) is expressed in terms of the resolvent, it decays both as n → ∞ and n → −∞ because n (cid:55)→ G E ( n ) is square summable. For | z | <
1, it follows from the textbelow (15) and similar arguments that there are solutions ˆΨ E + and ˆΨ E − with initial conditionsΨ E ( K + ) and Ψ E ( K − ) that decay at + ∞ and −∞ , respectively, in a square summable fashion.This implies that there are square matrices α z ± such that ˆΨ E + = Φ z + α z + and ˆΨ E − = Φ /z − α z − . Inparticular, Ψ E ( K + ) = Φ z + ( K + ) α z + , Ψ E ( K + ) = Φ /z − ( K + ) α z − . Because Ψ E ( K ± ) are of full rank L , the matrices α z ± are invertible. From the lower equationsof these identities, one deduces α z + = z − K + G E ( K + ) , α z − = z K − G E ( K − ) . Moreover,Φ /z − ( K + ) = T E ( K + , K − ) Φ /z − ( K − ) = T E ( K + , K − ) Ψ E ( K − ) ( α z − ) − = (cid:20) Ψ E ( K + ) − (cid:18) (cid:19)(cid:21) ( α z − ) − = (cid:20) Φ z + ( K + ) α z + − (cid:18) (cid:19)(cid:21) ( α z − ) − . Now using equations (21) and (35), one deduces M z − = − ν z Φ z + ( K + ) ∗ ı I Φ /z − ( K + )= − ν z Φ z + ( K + ) ∗ ı I (cid:20) Φ z + ( K + ) α z + − (cid:18) (cid:19)(cid:21) ( α z − ) − = 0 + ν z ı Φ z + ( K + ) ∗ (cid:18) (cid:19) ( α z − ) − = z K + z − z − ( α z − ) − = z K + − K − z − z − G E ( K − , K + ) − . Similarly N z − = ν z Φ /z + ( K + ) ∗ ı I Φ /z − ( K + )= ν z Φ /z + ( K + ) ∗ ı I (cid:20) Φ z + ( K + ) α z + − (cid:18) (cid:19)(cid:21) ( α z − ) − = α z + ( α z − ) − − ν z ı Φ /z + ( K + ) ∗ (cid:18) (cid:19) ( α z − ) − = z − K + − K − G E ( K + , K + ) G E ( K − , K + ) − − z − K + − K − z − z − G E ( K − , K + ) − . This shows the two identities. (cid:50) T z − and reflection matrix R z − interms of the Green matrix: T z − = z K − − K + ( z − z − ) G E ( K − , K + ) , R z − = z − K + (cid:0) − + ( z − z − ) G E ( K + , K + ) (cid:1) , (53)where | z | < z + z − / ∈ σ ( H ). The other coefficients of the scattering matrix for | z | < G E ( n, m ) ∗ = G E ( m, n ) . Due to T z + = ( T ¯ z − ) ∗ and the unitarity requirement, one finds T z + = z K − − K + ( z − z − ) G E ( K + , K − ) , R z + = z K − (cid:0) − ( z − z − ) G E ( K − , K − ) (cid:1) . (54)Hence the scattering matrix S z for | z | <
14 Examples If K + = K − so that V = 0, then there is no obstacle and the Jost solutions satisfy u z + = u z − .Then the transfer matrix as well as the scattering matrix is the identity. Next let us considerthe case of a perturbation on one site, say site 1. Thus K + = 1 and K − = 0. Then one dealsonly with M z (1 ,
0) = M z (1) given by (29) and, using the equations (4) and (43), one finds S z = (cid:32) (cid:0) − ı ν z V (1) (cid:1) − − ı ν z z − V (1) (cid:0) − ı ν z V (1) (cid:1) − − ı ν z z V (1) (cid:0) − ı ν z V (1) (cid:1) − (cid:0) − ı ν z V (1) (cid:1) − (cid:33) . Note that if | z | = 1, the inverse indeed always exists because the real part of − ı ν z V (1) ispositive. Further, if V (1) is of full rank, one obtains (12) in the limit z → ν z → ∞ . In general, let P be the projection onto the kernel of V (1), then T = P . Theasymptotically constant solution u φ of Proposition 27 can be chosen constant in this case.Therefore V has a reflectionless channel if P is non-trivial.Next let K + = 1 and K − = − M z (1 , −
1) = (cid:20) + ı ν z (cid:18) V (1) z − V (1) − z V (1) − V (1) (cid:19)(cid:21) (cid:20) + ı ν z (cid:18) V (0) V (0) − V (0) − V (0) (cid:19)(cid:21) . which allows to read off M z − = − ı ν z ( V (0) + V (1)) + ( ν z ) ( z − V (0) V (1)= − ı ν z ( V (0) + V (1) − zV (0) V (1))= − zz +1 V (0) V (1) − ı ν z ( V (0) + V (1) − V (0) V (1)) , ν z ( z −
1) = ız was used. Recalling that M z − is invertible for | z | = 1 (seeLemma 14), the transmission coefficient in this case is thus T z − = ( − ı ν z ( V (0) + V (1) − zV (0) V (1))) − . In the notations of Section 9, one can hence read off that G z = − zz +1 V (0) V (1) and F = − ı ( V (0) + V (1) − V (0) V (1)). It is now readily possible to construct examples withsingular F and hence non-generic band edge asymptotics. Acknowledgements:
The work of M. B. and G. F. C. was supported by PAPIIT IN108818and SEP-CONACYT 254062, that of H. S.-B. by PAPIIT-UNAM IN105718, CONACYTCiencia B´asica 283531 and the DFG.
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