{'fraction_non_alphanumeric': 0.06780675076033305, 'fraction_numerical': 0.033305080520516527, 'mean_word_length': 3.6204791522690623, 'pattern_counts': {'":': 0, '<': 26, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Darboux transformation for the discrete Schr\"odinger equation

The discrete Schr\"odinger equation on a half-line lattice with the Dirichlet boundary condition is considered when the potential is real valued, is summable, and has a finite first moment. The Darboux transformation formulas are derived from first principles showing how the potential and the wavefunction change when a bound state is added to or removed from the discrete spectrum of the corresponding Schr\"odinger operator without changing the continuous spectrum. This is done by explicitly evaluating the change in the spectral density when a bound state is added or removed and also by determining how the continuous part of the spectral density changes. The theory presented is illustrated with some explicit examples.


INTRODUCTION
Our goal in this paper is to analyze the Darboux transformation for the discrete Schrödinger equation on the half-line lattice with the Dirichlet boundary condition. In the Darboux transformation, the continuous part of the corresponding Schrödinger operator is unchanged and only the discrete part of the spectrum is changed by adding or removing a finite number of discrete eigenvalues to the spectrum. We can view the process of adding or removing discrete eigenvalues as changing the "unperturbed" potential and the "unperturbed" wavefunction into the "perturbed" potential and the "perturbed" wavefunction, respectively. Hence, our goal is to present the Darboux transformation formulas at the potential level and at the wavefunction level, by expressing the change in the potential and in the wavefunction in terms of quantities related to the perturbation and the unperturbed quantities.
The Darboux transformation was termed to honor the work of French mathematician Gaston Darboux [6], and it is useful for various reasons. For example, it allows us to produce explicit solutions to differential or difference equations by perturbing an already known explicit solution. As another example, we can mention that Darboux transformations for certain nonlinear partial differential equations or nonlinear partial differentialdifference equations yield so-called soliton solutions, which have important applications [12] in wave propagation of electromagnetic waves and surface water waves. We refer the reader to the existing literature [4,7,12,13,14] on the wide applications of Darboux transformation, and in our paper we concentrate on the mathematical aspects of the Darboux transformation for the Schrödinger equation on the half-line lattice with the Dirichlet boundary condition.
On the half-line lattice the discrete Schrödinger equation is given by −ψ n+1 + 2ψ n − ψ n−1 + V n ψ n = λ ψ n , n ≥ 1, (1.1) where λ is the spectral parameter, n is the spacial independent variable taking positive integer values, and the subscripts are used to denote the dependence on n. Thus, ψ n denotes the value of the wavefunction at n and V n denotes the value of the potential at n.
The point n = 0 corresponds to the boundary. We remark that (1.1) is the analog of the half-line Schrödinger equation (1.2) where λ is the spectral parameter, the prime denotes the x-derivative, ψ is the wavefunction, and V (x) is the potential. The point x = 0 corresponds to the boundary. In analogy to (1.2), we can use (1.1) to describe [15] the behavior of a quantum mechanical particle on a half-line lattice (such as a crystal) experiencing the force at each lattice point n resulting from the potential V n .
In order to determine the spectrum of the corresponding Schrödinger operator related to (1.1) and to identify a square-summable solution in n as an eigenfunction, we must impose a boundary condition on square-summable wavefunctions at n = 0. In applications related to quantum mechanics, it is appropriate to use the Dirichlet boundary condition at x = 0 for (1. The spectrum of the corresponding operator for (1.2) is well understood when the potential V (x) is real valued and satisfies the so-called L 1 1 -condition [4] given by Similarly, we assume that V n is real valued and satisfies the analog of (1.4) given by ∞ n=1 (1 + n) |V n | < +∞. (1.5) The class of real-valued potentials V (x) satisfying (1.4) is usually known [4] as the Faddeev class. Similarly, we refer to the set of real-valued potentials V n satisfying (1.5) as the Faddeev class. The existence of the first moments in (1.4) and (1.5) assures that the number of discrete eigenvalues for each of the corresponding Schrödinger operators is finite.
Our paper is organized as follows. In Section 2 we present the appropriate preliminaries involving the Jost solution and the regular solution to (1.1); the Schrödinger operator, the scattering states, the bound states, the Jost function, the scattering matrix, the phase shift, and the spectral density associated with (1.1) and ( The most relevant reference for our paper is [2], and in the current paper we use the notation used in [2]. The results in [2] were presented under the assumption that the potential is compactly supported, i.e. V n = 0 for n ≥ b for some positive integer b. In Section 2 we present the corresponding results when V n belongs to the Faddeev class and does not necessarily have a compact support. Another relevant reference for our paper is the classic work by Case and Kac [3]. Even though [3] is more related to the Jacobi operator and not to the Schrödinger operator, the treatment of the spectral density in [3] is useful. We remark that the Darboux transformation results related to the Jacobi operators do not reduce to the Darboux transformation results for the Schrödinger operator. Hence, in our paper we use the Gel'fand-Levitan theory [3,4,8] and an appropriate formula for the spectral density for the corresponding Schrödinger operator with bound states and derive the Darboux transformation from first principles.

PRELIMINARIES
In this section, associated with (1.1) and (1.3) we introduce various quantities such as the Jost solution f n , the regular solution ϕ n , the Jost function f 0 , the scattering matrix S, and the spectral measure dρ. We also present the basic properties of such quantities relevant to our analysis of Darboux transformations.  In quantum mechanics, it is customary to interpret the discrete spectrum associated with (1.1) and (1.3) as the bound states. Hence, the λ s -values in the discrete spectrum can be called the bound-state energies and the corresponding square-summable solutions can be called bound-state wavefunctions. The solutions to (1.1) when λ ∈ (0, 4) can be referred to as scattering solutions.
Associated with (1.1), instead of λ, it is convenient at times to use another spectral parameter related to λ, usually denoted by z, given by where the square root is used to denote the principal branch of the complex square-root function. Note that (2.1) yields Let us use T for the unit circle |z| = 1 in the complex plane C, T + for the upper portion of T given by z = e iθ with θ ∈ (0, π), and T + for the closure of T + given by z = e iθ with θ ∈ [0, π]. Under the transformation from λ ∈ C to z ∈ C, the real interval λ ∈ (0, 4) is mapped to z ∈ T + , the real half line λ ∈ (−∞, 0) is mapped to the real interval z ∈ (0, 1), the real interval λ ∈ (4, +∞) is mapped to the real interval z ∈ (−1, 0), the point λ = 0 is mapped to z = 1, and the point λ = 4 is mapped to z = −1. Using (2.2) it is convenient to write (1.1) as Let us now consider certain particular solutions to (1.1). A relevant solution to (1.1) or equivalently to (2.3) is the so-called regular solution ϕ n satisfying the initial conditions The result presented in the following theorem is already known and its proof is omitted.
A proof in our own notation can be obtained as in the proof of Theorem 2.6 of [2].
Theorem 2.1 Assume that the potential V belongs to the Faddeev class. Then, for n ≥ 1 the regular solution ϕ n to (1.1) with the initial values (2.4) is a polynomial in λ of degree n − 1 and is given by where, for each fixed positive integer n, the set of coefficients {B nj } n−1 j=0 are real valued and uniquely determined by the ordered set {V 1 , V 2 , . . . , V n−1 } of potential values. In particular, From (2.5) it is clear that the domain of ϕ n is the entire complex λ-plane. With the help of (2.2), we can conclude that the z-domain of ϕ n corresponds to the punctured complex z-plane with the point z = 0 removed.
Another relevant solution to (1.1) or equivalently to (2.3) is the Jost solution f n satisfying the asymptotic condition On the unit circle z ∈ T we have z −1 = z * , where we use an asterisk to denote complex conjugation. Let us use f n (z) to denote the value of f n when z ∈ T + . From (2.3) and (2.6) it follows that we have and hence the domain of f n (z) can be extended from z ∈ T + to z ∈ T by using (2.7).
We will see in Theorem 2.2 that, when the potential V n belongs to the Faddeev class, the domain of f n (z) can be extended from z ∈ T to the unit disc |z| ≤ 1.
Let us define g n as the quantity f n but by replacing z by z −1 there, i.e.
From (2.8) it follows that the domain of g n (z) is originally given as z ∈ T and it can be extended to |z| ≥ 1 when the potential V n in (1.1) belongs to the Faddeev class. With the help of (2.3) we see that g n is also a solution to (1.1), and from (2.6) it follows that g n satisfies the asymptotic condition The quantity f 0 , which is obtained from the Jost solution f n with n = 0, is known as the Jost function. Let us remark that the Jost solution f n is determined by the potential The Jost function f 0 (z) is used to define the scattering matrix S as (2.10) Even though S(z) is scalar valued, it is customary to refer to it as the scattering matrix.
With the help of (2.7) and (2.8) we see that we can write (2.10) in various equivalent forms such as Let us write the Jost function in the polar form as The real-valued quantity φ(z) appearing in (2.12) is usually called the phase shift. Its domain consists of z ∈ T. Using (2.7) in (2.12) we see that the phase shift satisfies where each coefficients K nm is real valued and uniquely determined by the potential values in the ordered set {V m } ∞ m=n+1 . In particular, we have The Jost function f 0 is analytic in z in |z| < 1 and continuous in z in |z| ≤ 1. It has the representation where each coefficient K 0m is uniquely determined by the set {V n } ∞ n=1 of potential values. In particular, we have (2.18) PROOF: It is enough to prove the analyticity in |z| < 1 and the continuity in |z| ≤ 1 for f n (z). The remaining results in (a)-(c) can be obtained with the help of Proposition 2.4 of [2]. Note that (2.18) is the same as (2.42) of [2] and the linear independence of f n and g n is established by using (2.6) and (2.9). Let us then prove the aforementioned analyticity and continuity. In fact, for the analyticity in |z| < 1, it is enough to use the summability in (1.5) without the need for the first moment of the potential. The first moment in (1.5) is needed to prove the continuity at z = ±1. We can prove the analyticity by modifying the proof of Lemma 1 of [7] so that it is applicable to the discrete Schrödinger equation.
We only provide the key steps and let the reader work out the details. Letting (2.20) Note that (2.20) is the discrete analog of the second displayed formula on p. 130 of [7].
Next we solve (2.20) iteratively by letting where we have defined n (z) is analytic in |z| < 1, and the left-hand side of (2.21) is analytic in |z| < 1 if we can show that the series on the right-hand side of (2.21) converges uniformly in every compact subset of |z| < 1. When |z| ≤ 1, we have The uniform convergence is established by using the estimates in (2.24) and (2.25). Hence, m n (z) is analytic in |z| < 1 for each fixed nonnegative integer n. From (2.19) it then follows that f n (z) is analytic in |z| < 1 for each fixed n ≥ 0. In order to prove the continuity of m n (z) in |z| ≤ 1, we need to show that each iterate m (p) n (z) is continuous in |z| ≤ 1 and that the series in (2.21) converges absolutely in |z| ≤ 1. The factor z −z −1 appearing in the denominator of (2.23) becomes troublesome at z = ±1. As a remedy, we use the identity With the help of (1.5), (2.22), (2.23), and (2.27), one establishes the uniform convergence in |z| ≤ 1 for the series on the right-hand side of (2.21). Furthermore, with the help of (2.23) and (2.26) we establish the continuity of each iterate m (p) n (z) in |z| ≤ 1. Then, it follows that m n (z) appearing on the left-hand side (2.21) is continuous in |z| ≤ 1. Finally, from (2.19) it follows that f n (z) is continuous in |z| ≤ 1 for each fixed value of n.
Let us remark that, from (2.16) and (2.17) we see that the value of the Jost function f 0 (z) at z = 0 is given by From the second equality of (2.15) it follows that The results in following theorem clarifies the generic and exceptional cases encountered at the endpoints of the continuous spectrum, i.e. at λ = 0 and λ = 4. (c) At λ = 4, or equivalently at z = −1, the regular solution ϕ n generically grows linearly in n as n → +∞, and in the exceptional case the regular solution ϕ n is bounded in n.
Hence, λ = 4 never corresponds to a bound state for (1.1) with the Dirichlet boundary condition (1.3). In the generic case we have f 0 = 0 at z = −1. In the exceptional case, f 0 has a simple zero at z = −1.
PROOF: The proofs (b) and (c) can be obtained as in the proof of Theorem 2.5 of [2].
The proof of (a) can be given as follows. Assume that f 0 (z) vanishes at some point z = z 0 , where z 0 is located on the unit circle T and z 0 = ±1. From (2.7) and (2.8) it follows that f 0 (z 0 ) = 0 implies that g 0 (z 0 ) = 0. Using these values in (2.18) we would then get ϕ n ≡ 0 for any positive integer n when z = z 0 . On the other hand, by the second equality in (2.4) we know that ϕ 1 must be equal to 1 when z = z 0 . This contradiction shows that f 0 cannot vanish on the unit circle, except perhaps at z = ±1.
In the next theorem, we summarize the facts relevant to the bound states of (1.  Let us assume that the bound states occur at λ = λ s for s = 1, . . . , N. Let us also assume that the corresponding z s -values are obtained via using (2.1), and hence the bound states occur at z = z s for s = 1, . . . , N. From (2.2) we see that From Theorem 2.4(b) we know that ϕ n (λ s ) is real valued and the quantity C s defined as we know that f n (z s ) is real valued and the quantity c s defined as is a finite nonzero number. It is appropriate to refer to the positive number c s as a Marchenko norming constant at z = z s . Thus, the quantity c s f n (z s ) is a normalized boundstate solution to (1.1) at the bound state z = z s . We then get Using the second equality of (2.4) in (2.31) we see that the Gel'fand-Levitan norming constant C s and the Marchenko norming constant c s are related to each other as Let us use a circle above a quantity to emphasize that it corresponds to the trivial potential V n ≡ 0 in (1.1). Hence, • ϕ n denotes the regular solution, • f n is the Jost solution, • g n is related to • f n as in (2.8), • f 0 is the Jost solution, and • S is the scattering matrix. We Let us use dρ to denote the spectral density corresponding to the Schrödinger equation its integral over the real-λ axis is equal to one. Let us use d • ρ to denote the spectral density when the potential is zero. From (4.1) of [2] we have where we recall that T + denotes the closure of the upper portion of the unit circle T.
In the absence of bound states, the spectral density dρ is given by where we recall that λ ∈ [0, 4] corresponds to z ∈ T + . Thus, the discrete part of the spectral density dρ is zero and the continuous part of the spectral density is obtained by where f 0 (z) is the corresponding Jost function and each z s corresponds to λ s via (2.28).
We remark that λ ∈ [0, 4] in (2.35) corresponds to z ∈ T + . Note that, in the absence of bound states, i.e. when N = 0, the spectral density given in (2.35) reduces to the expression given in (2.34). In the evaluation of (2.35) we have used the facts that With the help of (2.36) we see that the first line of (2.35) yields In order to understand the Darboux transformation, we need to establish the Gel'fand-Levitan formalism related to (1.1) and (1.3). Let V n andṼ n be the unperturbed and perturbed potentials, respectively. Let ϕ n andφ n be the respective corresponding regular solutions, and let dρ and dρ be the respective corresponding spectral densities. From where A nm are some real coefficients to be determined. Let us define the real-valued scalars G nm as We already have [2,3] the orthonormality λ∈R ϕ n dρ ϕ m = δ nm , (2.39) with δ nm denoting the Kronecker delta. Proceeding as in (4.13)-(4.17) of [2] we obtain the Gel'fand-Levitan system Analogous to (2.84) of [2], we get with the understanding that A 10 = 0.
(2.42) From (2.38) and (2.42) we see that G n−1 is a real symmetric matrix. For each integer n ≥ 2, we can write the Gel'fand-Levitan system (2.40) in the matrix form as where I n−1 is the (n − 1) × (n − 1) identity matrix. Let g n−1 be the column vector with (n − 1) components appearing on the right-hand side of (2.43), i.e.
Thus, A nm can be explicitly expressed in terms of the coefficients of G n−1 as where1 m is the column vector with (n − 1) components with all the entries being zero except for the mth entry being one. Note that the right-hand side of (2.46) contains a binomial form for a matrix inverse. Using (15) on p. 15 of [5], the binomial form in (2.46) can be expressed as a ratio of two determinants, yielding where the matrix in the numerator is a block matrix of size n × n. Using (2.47) in (2.37) and (2.41) we obtainφ n andṼ n in terms of the unperturbed quantities.
Let us refer to the data set {λ s , C s } N s=1 , which consists of all the bound-state energies and the corresponding Gel'fand-Levitan norming constants given in (2.29), as the boundstate data set. In general, the scattering matrix S(z) defined in (2.10) and the bound-state data set are independent. This is because the domain of S(z) consists of the unit circle z ∈ T and the bound-state energies correspond to the z s -values inside the unit circle. Let us consider the case where the nontrivial potential V n is compactly supported, i.e. when V n = 0 for n > b and V b = 0 for some positive integer b. Thus, we use b to signify the compact support of V n given by {1, 2, . . . , b}. For such potentials, it is known [2] that S(z) has a meromorphic extension from z ∈ T to the region |z| < 1 and the z s -values correspond to the poles of S(z) in |z| < 1. Furthermore, for such potentials the corresponding C s -values can be determined [2] in terms of certain residues evaluated at z s -values. In general, without a compact support, the values of z s and C s cannot be determined from the scattering matrix S(z). On the other hand, even without a compact support, when the potential V n belongs to the Faddeev class, the scattering matrix corresponding (1.1) and (1.3) contains some information related to the number of bound states N. This result is known as Levinson's theorem, and mathematically it can be viewed as an argument principle related to the integral of the logarithmic derivative of the scattering matrix along the unit circle T in the complex z-plane.
In the next theorem, we present Levinson's theorem associated with (1 .1) and (1.3).
For this purpose it is appropriate to introduce the constants µ + and µ − as Let ∆ T acting on a function of z denote the change in that function when the z-value moves along the unit circle T once in the counterclockwise direction in the sense of the Cauchy principal value. By the sense of the Cauchy principal value, we mean that in the evaluation of the change by using an integral along T, we interpret the corresponding integral as a Cauchy principal value. In the theorem given below, that amounts to integrating along the unit circle z = e iθ for θ ∈ (0 + , π − 0 + ) ∪ (π + 0 + , 2π − 0 + ) because the only singularities for the integrand may occur at z = 1 or z = −1. (a) The change in the phase shift φ(z) when z moves along T in the counterclockwise direction once is given by
(b) The change in the phase shift φ(z) when z moves along T + from z = 1 to z = −1 is given by The change in the argument of S(z) when z moves along T + from z = 1 to z = −1 is given by The change in the argument of f 0 (z) when z moves along T + from z = 1 to z = −1 is given by to f 0 (z) along the unit circle, we see that the change in the argument of f 0 (z) as z moves along the unit circle once in the counterclockwise direction is given by

DARBOUX TRANSFORMATION IN ADDING A BOUND STATE
In this section we determine the effect of adding a bound state to the discrete spectrum of the Schrödinger operator corresponding to (1.1) and (1.3). For clarity, we use the notation V n (N ) for V n to indicate that the Schrödinger operator contains exactly N bound states occurring at λ = λ s for s = 1, . . . , N. Hence, we order the bound states by assuming that we start with the potential V n (0) containing no bound states. Then, we add one bound state at λ = λ 1 with some Gel'fand-Levitan norming constant and obtain the potential V n (1). Next, we add one bound state at λ = λ 2 with some Gel'fand-Levitan norming constant and obtain the potential V n (2). Continuing in this manner we recursively add all the bound states with λ = λ s for s = 1, . . . , N and obtain the potential V n (N ). Note that (2.28) establishes a one-to-one correspondence between λ s and z s , and hence we can equivalently say that the bound states of the potential V n (N ) occur at z = z s for s = 1, . . . , N. We remark that the ordering of λ s is completely arbitrary and that ordering does not have to have λ s in an ascending or descending order.
To the "unperturbed" potential V n (N ) let us add one bound state at λ = λ N+1 with the Gel'fand-Levitan norming constant C N+1 . We then get the "perturbed" potential in the continuous spectrum remains unchanged [4]. However, this is no longer the case for the discrete Schrödinger equation. Let us elaborate on this matter. We would like f 0 (z; N + 1) to be obtained from f 0 (z; N ) via where Q(z) is analytic in |z| < 1, continuous in |z| ≤ 1, and satisfies Q(0) = 1. The constraints on Q(z) are determined by the fact that both f 0 (z; N + 1) and f 0 (z; N ) must be analytic in |z| < 1, continuous in |z| ≤ 1, and take the value of 1 at z = 0, as required by Theorem 2.2(b). Furthermore, f 0 (z; N + 1) must have a simple zero at z = z N+1 and f 0 (z; N ) must be nonzero when z = z N+1 . The further requirement |f 0 (z; N + 1)| = |f 0 (z; N )|, z ∈ T, (3.2) combined with the maximum modulus principle would yield In establishing a Darboux transformation, the choice of Q(z) appearing in (3.1) is not unique. We find it convenient to choose a particular Q(z) as which greatly simplifies evaluations involving the spectral density given in (2.35). On the other hand, the choice Q(z) ≡ 1 yields which hinders evaluations involving the spectral density. Another advantage of the choice of Q(z) given in (3.4) is that the pole of Q(z) at z = 1/z N+1 can be considered as a real-valued resonance for the discrete Schrödinger equation (1.1), where we recall that z N+1 ∈ (−1, 0) ∪ (0, 1). Consider the special case of a compactly-supported potential, where z = z N+1 is a real-valued resonance for V n (N ), i.e. f 0 (z; N ) has a simple zero at z = 1/z N+1 . We may then be able to convert that resonance to a bound state by adding a bound state to V n (N ) at z = z N+1 in such a way that V n (N + 1) contains a bound state.
We refer the reader to [1], where the analogous problem for (1. Let S(z; N ) and S(z; N + 1) denote the scattering matrices for the unperturbed and perturbed problems, respectively. From (2.11) we get Using (3.6) in (3.7), after some simplification, we obtain the Darboux transformation for the scattering matrix as One can directly verify that and hence, with the help of (2.14), we see that the Darboux transformation for the phase shift is given by Next, let us determine the Darboux transformation for the spectral density. Let dρ(λ; N ) and dρ(λ; N + 1) denote the unperturbed and perturbed spectral densities, respectively. From (2.35) we see that where we recall that λ ∈ [0, 4] corresponds to z ∈ T + . Using (3.5) in (3.11) we see that and hence from (3.10) and (3.12) we get the Darboux transformation for the spectral density as (3.13) Our next goal is to determine the Darboux transformation for the regular solution. In other words, we would like to determine the relationship between ϕ n (λ; N ) and ϕ n (λ; N + 1), where the former is the regular solution for the unperturbed problem and the latter is the regular solution for the perturbed problem.
Let us now use the Gel'fand-Levitan procedure in the special case with V n (N + 1) denotingṼ n and V n (N ) denoting V n . In that special case dρ and dρ appearing in (2.38) correspond to dρ(λ; N ) and dρ(λ; N + 1), respectively, appearing on the left-hand side of (3.13). The unperturbed and perturbed regular solutions ϕ n andφ n appearing in (2.37) correspond to ϕ n (λ; N ) and ϕ n (λ; N + 1), respectively. From the second line of (3.10) we With the help of (2.39) and (3.14) we get where we recall that δ nm denotes the Kronecker delta. Using (3.13) in (2.38) we obtain 4] ϕ n (λ; N ) dρ(λ; N ) ϕ m (λ; N ) (3.16) The integral in (3.16) is equal to the right-hand side of (3.15). Thus, from (3.15) and (3.16) we obtain (3.17) Having obtained G nm as in (3.17) in terms of the unperturbed quantities related to V n (N ), one can then use G nm in (2.37) and (2.41) in (2.45) in order to determine ϕ n (λ; N + 1) and V n (N + 1), respectively.
Alternatively, in order to obtain ϕ n (λ; N + 1) and V n (N + 1), we can proceed as follows. Let us write (3.17) in terms of the real-valued (N + 1) × (N + 1) diagonal matrix E N and the real-valued column vector ξ n with N + 1 entries as where we have defined We recall that the dagger in (3.20) can also be replaced by the matrix transpose since the column vector ξ n is real valued. From (3.18) we see that G nm is separable in n and m.
Thus, we can solve the Gel'fand-Levitan system (2.40) explicitly by seeking A nm in the form A nm = β † n ξ m , 1 ≤ m < n, (3.21) where the column vector β n has N +1 components that are to be determined. Using (3.18) and (3.21) in (2.40) we observe that β † n satisfies where we recall that I N+1 denotes the (N + 1) × (N + 1) identity matrix. From (3.22) we obtain , n = 2, 3, . . . , (3.23) which simplifies to From (3.21) and (3.24) we see that Hence, for n ≥ 2, from (2.41) and (3.25) we obtain the Darboux transformation at the potential level as (3.26) Since A 10 = 0, for n = 1, instead of (3.26) we need to use which is obtained from (3.26) by replacing the first term on the right-hand side by zero and by using n = 1 in the second term. Note that ξ 1 ξ † 1 appearing in (3.27) is the (N +1)×(N +1) matrix with all entries being equal to one.
Let us remark that (3.25)-(3.27) contain some binomial forms for the inverse of a matrix. Using (15) on p. 15 of [5], such binomial forms can be expressed as a ratio of two determinants. For example, we can write the right-hand side of (3.25) as where we have defined num as the determinant of the (N + 2) × (N + 2) block matrix given by num := det and we have defined den as the determinant of the (N + 1) × (N + 1) matrix given by As the next proposition shows, the summation term in (3.31) can be written as a linear combination of ϕ n−1 (λ; N ) and ϕ n (λ; N ). Let us define the real-valued column vector α n (λ) with N + 1 components as where α n (λ) is the real-valued column vector defined in (3.32) with N + 1 components.
(b) The (N + 1) × (N + 1) matrix consisting of the summation term in (3.24) can be simplified and its (k, l)-component for n ≥ 2 is given by where the overdot denotes the λ-derivative. Thus, the proof of (a) is complete. Let us now turn the proof of (b). From (3.20) and the fact that ξ j is real, we see that the (k, l)-component of ξ j ξ † j is given by From (3.37) and (3.38) we see that, when k = l, we have yielding the first line of (3.34). When k = l, we can use the limit λ → λ s in (3.37), which gives us yielding the second line of (3.34).
Using where we recall that β † n is the real-valued row vector in (3.24), α n (λ) is the real-valued column vector given in (3.32), and ξ n is the real-valued column vector given in (3.20).

DARBOUX TRANSFORMATION IN REMOVING A BOUND STATE
In this section we determine the effect of removing a bound state from the discrete spectrum of the Schrödinger operator corresponding to (1. 1) and (1.3). For clarity, we use the notation introduced in Section 3. We have the unperturbed potential V n (N ) containing N bound states at λ = λ s for s = 1, . . . , N. We then remove the bound state at λ = λ N with the Gel'fand-Levitan norming constant C N in order to obtain the perturbed potential V n (N − 1) containing N − 1 bound states. As in Section 3, we know from (2.28) that there is a one-to-one correspondence between λ s and z s , and hence we can equivalently say that the bound states of the potential V n (N ) occur at z = z s for s = 1, . . . , N, and we remove the bound state at z = z N .
The Darboux transformation for the Jost solution in going from f 0 (z; N ) to f 0 (z; N −1) can be obtained via (3.6) as Similarly, the Darboux transformation for the scattering matrix in going from S(z; N ) to S(z; N − 1) can be obtained via (3.8) as With the help of (3.9) we see that the Darboux transformation for the phase shift in going from φ(z; N ) to φ(z; N − 1) can be obtained via (3.9) as Let us now determine the Darboux transformation for the spectral density in going from dρ(λ; N ) to dρ(λ; N − 1). From (3.10) we see that On the other hand, from (3.5) we have (4.4) We recall that λ ∈ [0, 4] in (4.2) and (4.4) corresponds to z ∈ T + . Thus, from (3.10) and (4.4) we get where the constants A nm are to be determined from (2.40) by using (4.6) as input. In this case, from (2.41) we get again with the understanding that A 10 = 0. Using (4.5) in (4.6) we obtain (4.7) Using (3.15) in (4.7), after some simplification we get (4.8) Proceeding as in (3.18)-(3.20) we can write G nm given in (4.8) as 9) where F N is the N × N diagonal matrix with real entries given by Comparing (3.20) and (4.11) we observe that the first N entries of the column vectors θ n and ξ n are identical and that ξ n has an additional (N + 1)st entry. As in Section 3, the quantity G nm given in (4.9) is separable in n and m, and hence the Gel'fand-Levitan system (2.40) is explicitly solvable by using the analog of (3.21), i.e. by letting where the column vector γ n has N components to be determined. Proceeding as in (3.22)-(3.25) we determine γ † n as (4.13) From (4.12) and (4.13) we see that (4.14) The analogs of (3.28)-(3.30) also apply in this case. Since the right-hand side of (4.12) is a binomial for a matrix inverse, we can write A nm given in (4.12) as the ratio of two determinants as As in Proposition 3.1(b), for n ≥ 2 we can simplify the N × N matrix n−1 j=1 θ j θ † j appearing in (4.13)-(4.15) and find that its (k, l)-entry is given by The analog of (3.26) in this case is obtained by using (4.14) in (2.41), and for n ≥ 2 we get the Darboux transformation in going from V n (N ) to V n (N − 1) given by (4.17) For n = 1, instead of (4.17) we use the analog of (3.27) and get where ǫ n (λ) is the column vector with N components and it is defined as We remark that the column vector ǫ n (λ) given in (4.19) has N components, and the column vector α n (λ) given in (3.32) has N + 1 components. In fact, ǫ n (λ) is obtained from α n (λ) by omitting the last entry.

SOME EXPLICIT EXAMPLES
In this section we illustrate the results of the previous sections with some explicit examples. We also make some contrasts between the Darboux transformation for (1.1) and the Darboux transformation for (1.2) when the potentials are compactly supported.
Let us consider the case where the potential V n in (1.1) is nontrivial and compactly supported, i.e. assume that V n = 0 for n > b and V b = 0 for some positive integer b.
The corresponding Jost function f 0 appearing in (2.10) is then a polynomial in z of degree 2b − 1 and, as (2.50) of [2] indicates, is given by for some nonzero constants α 1 , α 2 , α 3 satisfying In case the system (5.3) is inconsistent, the quantity given on the right-hand side of (5.4) cannot be the Jost solution of a compactly-supported potential.
For the half-line Schrödinger equation (1.2) with a compactly-supported potential V (x), the following property is known [1]. If we remove a bound state from such a potential, then the transformed potential is also compactly supported and the transformed potential is guaranteed to vanish outside the support of the original potential. In some of the examples in this section, we illustrate that the aforementioned support property does not necessarily hold for the discrete Schrödinger equation (1.1). We show that the property holds in one example but does not hold in another example.
For the half-line Schrödinger equation (1.2) with a compactly-supported potential V (x), also the following second property holds [1]. If we add a bound state to a compactlysupported potential, then the transformed potential is also compactly supported (and the transformed potential is guaranteed to vanish outside the support of the original potential) if and only if the two conditions specified in Theorem 3.5 of [1] are satisfied. The first condition is that the added bound-state λ s -value must come from an "eligible" resonance [1] and the second condition is that the corresponding Gel'fand-Levitan norming constant C s must have a specific positive value. In some of the examples in this section, we illustrate that the aforementioned support property does not necessarily hold for the discrete Schrödinger equation (1.1). We show that the property holds in one example but does not hold in another example.
In the next example, we add a bound state at z = z 1 with the Gel'fand-Levitan norming constant C 1 to a compactly-supported potential with b = 1. The example shows that the Darboux transformation on the compactly-supported potential results in a compactlysupported potential if the values for z 1 and C 1 are chosen appropriately.
Example 5.1 Consider the compactly-supported potential V n with b = 1 and hence V n = 0 for n ≥ 2. Let us assume that 0 < |V 1 | ≤ 1. From (5.1) we see that the Jost function is given by Using (2.4) in (2.3), we obtain the corresponding regular solution ϕ n as a function of z as Since the bound states correspond to the zeros of f 0 when z ∈ (−1, 0) ∪ (0, 1), from (5. 6) we see that there are no bound states and hence we have N = 0. Let us now add one bound state at z = z 1 with the Gel'fand-Levitan norming constant C 1 . Let us choose z 1 = −V 1 , and hence impose the further restriction 0 < |V 1 | < 1. Let us usef 0 andṼ n to denote the corresponding Jost function and potential, respectively, when the bound state is added.
From (3.6) and (5.6) we see thatf The quantity E N defined in (3.19) with N = 0 is given by E 0 = C 2 1 . Then, (3.27) and (3.26) respectively yieldṼ From (5.10) we see thatṼ n is compactly supported if and only if we have In fact, with the special choice of the Gel'fand-Levitan norming constant in (5.11), from (5.9) we obtainṼ 1 = 1/V 1 . In the presence of one bound state for the compactly-supported potentialṼ n , the corresponding Gel'fand-Levitan norming constant C 1 can be evaluated with the help of (2.32), (5.2), (5.8), and the fact thatf 1 = z, yielding the value of C 2 1 given in (5.11).
In the following example, we illustrate that a polynomial in z of degree 2b − 1 may or may not correspond to the Jost function of a compactly-supported potential.
Example 5.2 Consider the Jost function f 0 given by Comparing (5.2) with (5.1), we see that one solution to the corresponding system (5.2) From (5.2) we see that f 0 has two zeros when z ∈ (−1, 0) ∪ (0, 1), and hence it has two bound-state zeros given by z 1 = −1/2 and z 2 = 1/2. From (2.36) we see that the corresponding Gel'fand-Levitan norming constants C 1 and C 2 must satisfy 0 < C 2 1 + C 2 2 ≤ 1. Corresponding to a compactly-supported potential we must [2] have f n = z n for n ≥ b.
Hence, in our example, corresponding to (5.6) we have f 2 = z 2 and f 3 = z 3 . Then, from where the overline on a digit indicates a round off. We note that (5.13) is compatible with the constraint 0 < C 2 1 + C 2 2 ≤ 1. Thus, we have confirmed that z 1 = −1/2 and z 2 = 1/2 do indeed correspond to bound states of the compactly-supported potential described in (5.12). In (5.4), if we choose α j = 1 for j = 1, 2, 3, then the system in (5. We remark that it is impossible to have a compactly-supported potential with b = 2 having three bound states. This can be seen as follows. Assume that for some choice of V 1 and V 2 in (5.3) we had −1 < α 1 < α 2 < α 3 < 1 for nonzero α j values. Using (5.4) in (2.10) and (5.2) we would get the corresponding Marchenko norming constants as . (5.14) From (5.14) we see that we would have c 2 1 > 0, c 2 2 < 0, c 2 3 > 0, and hence it is impossible to have N = 3. From Example 5.1 we know that 0 ≤ N ≤ b when b = 1, and from (5.14) we know that 0 ≤ N ≤ b when b = 2. From (5.1) it is clear that the number of zeros of f 0 (z) in z ∈ (−1, 0) ∪ (0, 1) cannot exceed 2b − 1. We pose the following as an open problem, which can perhaps be answered with the help of a generalization of (5.14) from b = 2 to an arbitrary positive integer b : For any given positive integer b, what is the maximal number of bound states for the corresponding Schrödinger operator associated with (1.1) and (1.4), if the potential V n has a compact support with V n = 0 for n > b?

(5.17)
In the next two examples, we show that if we remove a bound state from a compactlysupported potential then the resulting potential may or may not be compactly supported.

Example 5.3
Consider the compactly-supported potential V n with b = 1 and hence V n = 0 for n ≥ 2. The corresponding Jost function is given by (5.6). Since the bound states correspond to the zeros of f 0 when z ∈ (−1, 0) ∪ (0, 1), from (5.6) we see that there exists one bound state if |V 1 | > 1. We assume that |V 1 | > 1 so that we have exactly one bound state at z = z 1 , where z 1 = −1/V 1 . From (2.10) and (5.6) we see that the corresponding scattering matrix is given by In this case, the Jost solution satisfies f n = z n for n ≥ 1. In the presence of one bound state, the corresponding Gel'fand-Levitan norming constant C 1 is evaluated with the help of (2.32), (5.2), (5.18), and f 1 = z, yielding From (2.36) we see that we must have 0 < C 2 1 ≤ 1 and hence we must use the restriction 0 < |V 1 | ≤ √ 2. Let us now remove the bound state with z 1 = −1/V 1 . The transformed Jost solutionf 0 is obtained via (4.1) and is given byf 0 = 1 + z/V 1 . In this case, using Using (5.19) with N = 1, we get the quantity F N given in (4.10) as Using (5.20) and (5.21) in (4.17) and (4.18) we obtainṼ n = 0 for n ≥ 2 andṼ 1 = 1/V 1 .
Example 5.4 Consider the compactly-supported potential V n described by (5.12) in Example 5.2. We know from Example 5.2 that there are two bound states with z 1 = −1/2 and z 2 = 1/2 with the respective corresponding Gel'fand-Levitan norming constants C 1 and C 2 as in (5.13). Hence, we have N = 2. We now demonstrate that if we remove the bound state at z = z 2 by using the Darboux transformation formulas given in Section 4 then the transformed potential is no longer compactly supported. From (2.28) we see that the values λ 1 and λ 2 corresponding z 1 and z 2 , respectively, are given by With the help of (5.13), (5.23), and (5.24), from (4.17) and (4.18) we can evaluate the transformed potentialṼ n for all n ≥ 1. We list the first few values below and mention that V n is not compactly supported: .