Continuous and discrete dynamical Schr\"odinger systems: explicit solutions

TL;DR

This paper develops explicit solutions for continuous and discrete dynamical Schr"odinger systems with matrix potentials using a generalized Bäcklund-Darboux transformation, and explores their asymptotic behavior over time.

Contribution

It introduces a generalized Bäcklund-Darboux transformation method to construct explicit solutions for dynamical Schr"odinger systems with matrix potentials.

Findings

Constructed explicit solutions for dynamical Schr"odinger systems.

Analyzed asymptotic expansions of solutions in time.

Extended the Bäcklund-Darboux transformation to a generalized form.

Abstract

We consider continuous and discrete Schr\"odinger systems with self-adjoint matrix potentials and with additional dependence on time (i.e., dynamical Schr\"odinger systems). Transformed and explicit solutions are constructed using our generalized (GBDT) version of the B\"acklund-Darboux transformation. Asymptotic expansions of these solutions in time are of interest.

Equations

$$\displaystyle\mathrm{i}\frac{\partial}{\partial t}\psi(x,t)=\big{(}H\psi\big{)}(x,t),\quad H:=-\frac{\partial^{2}}{\partial x^{2}}+u(x)\quad(u=u^{*}),$$

$$\mathrm{i}\left(\frac{\partial }{\partial t}\Psi \right)(t)=\mathcal{J}\Psi (t),$$

$$\mathcal{J}=\left[\begin{array}{cccccc}{b}_1& a_1& 0& 0& 0& \dots \\ c_2& b_2& a_2& 0& 0& \dots \\ 0& c_3& b_3& a_3& 0& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right],\Psi (t)=\left[\begin{array}{c}{\psi }_1(t)\\ {\psi }_2(t)\\ {\psi }_3(t)\\ \dots \end{array}\right].$$

$$-y^{\prime \prime }(x,\lambda )+u(x)y(x,\lambda )=\lambda y(x,\lambda )(u=u^{\ast }),y^{\prime }:=\frac{d}{dx}y.$$

$$w^{\prime }(x,\lambda )=G(x,\lambda )w(x,\lambda ),G(x,\lambda )=-\lambda q_1(x)-q_0(x),$$

$$q_{1}=\left[\begin{array}[]{lr}0&0\\ I_{h}&0\end{array}\right],\hskip 10.00002ptq_{0}(x)=-\left[\begin{array}[]{lr}0&I_{h}\\ u(x)&0\end{array}\right],\quad u(x)=u(x)^{*},$$

$$q_k^{\ast }=-j{q}_k{j}^{\ast }(k=0,1),j:=\left[\begin{array}{cc}0& I_h\\ -I_h& 0\end{array}\right],j^{\ast }=j^{-1}=-j.$$

$$AS(0)-S(0)A^{\ast }=\Pi (0)j\Pi (0{)}^{\ast }.$$

$${\Pi }^{\prime }=A\Pi q_1+\Pi q_0,S^{\prime }=\Pi q_{1}j{\Pi }^{\ast },$$

$$S(x)=S(x{)}^{\ast }.$$

$$AS(x)-S(x)A^{\ast }=\Pi (x)j\Pi (x{)}^{\ast }$$

$$w_A(x,\lambda )=I_m-j\Pi (x{)}^{\ast }S(x{)}^{-1}(A-\lambda I_n{)}^{-1}\Pi (x),$$

$$w_A^{\prime }(x,\lambda )=\tilde{G}(x,\lambda )w_A(x,\lambda )-w_A(x,\lambda )G(x,\lambda ),$$

$$\tilde{G}(x,\lambda ):=-\lambda q_1(x)-{\tilde{q}}_0(x),$$

$${\tilde{q}}_0=q_0-(q_{1}jX-jX{q}_1),X(x):=\Pi (x{)}^{\ast }S(x{)}^{-1}\Pi (x).$$

$$q_{1}j=\left[\begin{array}{cc}0& 0\\ 0& I_h\end{array}\right]\ge 0,S^{\prime }(x)\ge 0,$$

$$({\Pi }^{\ast }{S}^{-1}{)}^{\prime }(x)=q_1^{\ast }\Pi (x{)}^{\ast }S(x{)}^{-1}A+{\tilde{q}}_0(x{)}^{\ast }\Pi (x{)}^{\ast }S(x{)}^{-1}.$$

$$X(x)=X(x{)}^{\ast }.$$

$$\Pi =\left[\begin{array}{cc}{\Phi }_1& {\Phi }_2\end{array}\right],X=\{X_{ij}{\}}_{i,j=1}^2,X_{ij}={\Phi }_i^{\ast }{S}^{-1}{\Phi }_j.$$

$${\tilde{q}}_0=-\left[\begin{array}{cc}-X_{22}& I_h\\ u+X_{12}+X_{21}& X_{22}\end{array}\right].$$

$$\displaystyle Z=\begin{bmatrix}z_{1}\\ z_{2}\end{bmatrix}:=\Pi^{*}S^{-1},\quad q_{1}^{*}=\left[\begin{array}[]{lr}0&I_{h}\\ 0&0\end{array}\right],$$

$${\tilde{q}}_0^{\ast }=-\left[\begin{array}{cc}-X_{22}& u+X_{12}+X_{21}\\ I_h& X_{22}\end{array}\right],$$

$$z_1^{\prime }=z_{2}A+X_{22}{z}_1-(u+X_{12}+X_{21})z_2,z_2^{\prime }=-z_1-X_{22}{z}_2.$$

$$z_2^{\prime \prime }=-z_{2}A+(u+X_{12}+X_{21}-X_{22}^{\prime }+X_{22}^2)z_2.$$

$$\psi (x,t)=\left[\begin{array}{cc}0& I_h\end{array}\right]\Pi (x{)}^{\ast }S(x{)}^{-1}{\mathrm{e}}^{-\mathrm{i}tA}$$

$$\displaystyle\mathrm{i}\frac{\partial}{\partial t}\psi(x,t)=\big{(}\widetilde{H}\psi\big{)}(x,t),\quad\widetilde{H}:=-\frac{\partial^{2}}{\partial x^{2}}+\widetilde{u}(x),$$

$$\widetilde{u}(x)=u(x)+2\big{(}X_{12}(x)+X_{21}(x)+X_{22}(x)^{2}\big{)}.$$

$$X_{22}^{\prime }$$

$$=\left[\begin{array}{cc}0& I_h\end{array}\right]{\tilde{q}}_0^{\ast }{\Pi }^{\ast }{S}^{-1}\Pi \left[\begin{array}{c}0\\ I_h\end{array}\right]-\left[\begin{array}{cc}0& I_h\end{array}\right]{\Pi }^{\ast }{S}^{-1}\Pi \left[\begin{array}{c}{I}_h\\ 0\end{array}\right]$$

$$=-X_{12}-X_{22}^2-X_{21}.$$

$$z_2(x)A$$

$$=-z_2^{\prime \prime }(x)+\tilde{u}(x)z_2(x).$$

$$\displaystyle\big{(}S(x)^{-1}\big{)}^{\prime}=-S(x)^{-1}\Pi(x)\begin{bmatrix}0&0\\ 0&I_{h}\end{bmatrix}\Pi(x)^{*}S(x)^{-1}.$$

Full text

MSC(2010): 35Q41, 37C80, 39A12, 47B36.

Keywords: *Schrödinger equation, dynamical system, Jacobi matrix,

Bäcklund-Darboux transformation, dispersion, explicit solution.*

1 Introduction

Dynamical Dirac and Schrödinger systems play an essential role in mathematical physics and are actively studied, especially in the recent years (see, e.g., [2, 3, 4, 7, 8, 17, 21, 27, 28, 35] and numerous references therein). Continuous dynamical Schrödinger system has the form:

[TABLE]

where $u$ is an $h\times h$ matrix function, $h\in{\mathbb{N}}$, and ${\mathbb{N}}$ is the set of natural numbers. The matrix function $u$ is called the potential of (1.1) and this potential does not depend on $t$ in our case.

In discrete dynamical Schrödinger system we use Jacobi matrices ${\mathcal{J}}$ instead of $H$ since Jacobi operators “can be viewed as the discrete analogue of Sturm-Liouville operators” [34, Preface]. The corresponding system is given by the formula:

[TABLE]

where ${\mathcal{J}}$ is a semi-infinite block Jacobi matrix and $\Psi$ is a block vector

[TABLE]

Here, the blocks $a_{k}$, $b_{k}$ and $c_{k}$ are $h\times h$ matrices and $c_{k+1}=a_{k}^{*}$ $(k\geq 1)$.

Explicit solutions of dynamical systems are important as models and examples and they are also essential in applications. Various explicit solutions of time-independent systems were constructed using commutation methods [6, 11, 12, 18] and several versions of Bäcklund-Darboux transformations. Bäcklund-Darboux transformation is a well-known tool in the spectral theory and theory of explicit solutions. The original equation, which was studied by Darboux, is the Schrödinger equation

[TABLE]

Later, and especially in the last 40 years, this transformation was greatly modified, generalized and applied to a variety of linear and nonlinear equations (see, e.g., [5, 14, 19, 20, 30]).

It was shown recently in [28, 29] that the GBDT version of Bäcklund-Darboux transformation (for GBDT see [22, 23, 24, 25, 26, 30] and references therein) may be successfully applied to the construction of explicit solutions of dynamical systems as well.

In the present paper, we consider the important case of continuous dynamical Schrödinger system and a more difficult case of discrete system (i.e., system (1.2)). Some preliminaries are presented in Section 2, continuous dynamical Schrödinger system is dealt with in Section 3 and system (1.2) is considered in Section 4.

The dependence of our solutions of (1.1) and (1.2) on time is described by the factor $\mathrm{e}^{\mathrm{i}tA}$, where $A$ is a parameter matrix (generalized eigenvalue) of the GBDT transformation. Since $A$ is not necessarily self-adjoint and may have Jordan cells of different orders, the asymptotic expansion of our solutions of (1.1) and (1.2) essentially differs (see Remark 3.4) from the classical Jensen-Kato formulas (see [15] as well as further references in [7, 8]).

As usual, ${\mathbb{R}}$ denotes the set of real values, ${\mathbb{N}}$ is the set of natural numbers, and the complex plane is denoted by ${\mathbb{C}}$. Notation $\mathcal{S}^{*}$ stands for the matrix which is the conjugate transpose of $\mathcal{S}$, we write $\mathcal{S}>0$ when $\mathcal{S}$ is a positive-definite matrix, and $I_{m}$ stands for the $m\times m$ identity matrix. The notation $J=\mathrm{diag}\{J_{1},J_{2},\ldots\}$ means that $J$ is a diagonal or block diagonal matrix with the entries (or block entries) $J_{1}$, $J_{2}$ and so on.

2 GBDT: preliminaries

GBDT (generalized Bäcklund-Darboux transformation) was first introduced in [22], and a more general version of GBDT for first order systems rationally depending on the spectral parameter (in particular, for systems of the form $w^{\prime}=G(x,\lambda)w$, $G(x,\lambda)=-\sum_{k=-r}^{r}\lambda^{k}q_{k}(x)$) was treated in [23, 30] (see also some references therein). First order system

[TABLE]

where $w$ takes values in ${\mathbb{C}}^{m}$ $(m:=2h)$ and $m\times m$ coefficients $q_{1}$ and $q_{2}$ have the form

[TABLE]

is equivalent to the matrix Schrödinger equation (1.4) with a self-adjoint $h\times h$ potential $u(x)$. Here we present basic GBDT results for this system (see, e.g., [23, 24]). The connection with the Schrödinger equations (1.1) and (1.4) is discussed in greater detail in the next section.

Remark 2.1

We consider systems (2.1) and (1.1) on finite or infinite intervals ${\mathcal{I}}$, that is, we assume that $x\in{\mathcal{I}}$. Without loss of generality we assume also that $0\in{\mathcal{I}}$ and speak later about parameter matrices $S(0)$ and $\Pi(0)$ instead of $S(x_{0})$ and $\Pi(x_{0})$ for some fixed $x_{0}\in{\mathcal{I}}$. The most interesting for us is the case of the semiaxis ${\mathcal{I}}=[0,\infty)$.

In general, GBDT is determined by the choice of 5 parameter matrices (this case was treated in [24], where $u$ was not necessarily self-adjoint). However, relations (2.2) (including $u=u^{*}$) imply additional equalities:

[TABLE]

Thus the conditions of Proposition 1.4 from [23] are fulfilled, and we may use this proposition and some formulas from its proof. The following statements in this section are particular cases of [24, Theorem 2.1] (or [23, Theorem 1.2]) completed by [23, Proposition 1.4].

Hence, in the present case we use 3 parameter matrices. More precisely, we choose some initial system (2.1) (or, equivalently, the initial potential $u=u^{*}$ of Schrödinger equation (1.4)) and fix $n\in{\mathbb{N}}$. Then, we fix $n\times n$ matrices $A$ and $S(0)=S(0)^{*}$, and an $n\times m$ $(m=2h)$ matrix $\Pi(0)$ such that the following matrix identity holds:

[TABLE]

Suppose that such parameter matrices are fixed and that the potential $u(x)$ is locally summable on ${\mathbb{R}}$. Now, we can introduce matrix functions $\Pi(x)$ and $S(x)$ with the values $\Pi(0)$ and $S(0)$ at $x=0$ as the solutions of the linear differential equations

[TABLE]

where $q_{1}$ and $q_{0}$ are given by (2.2), and so $q_{1}j=(q_{1}j)^{*}$. Thus, in view of $S(0)=S(0)^{*}$, we have

[TABLE]

Notice that equations (2.5) are constructed in such a way that the identity

[TABLE]

follows (for all $x\in{\mathbb{R}}$) from (2.4) and (2.5). (The relation is obtained by the direct differentiation of the both sides of (2.7).) Assuming that $\det S(x)\not\equiv 0$ we can define a matrix function

[TABLE]

where $\lambda\not\in\sigma(A)$ ($\sigma$ means spectrum).

Theorem 2.2

Suppose that the relation (2.4) is valid, and matrix functions $\Pi(x)$ and $S(x)$ satisfy equations (2.5) where (2.2) holds. Then, in the points of invertibility of $S(x)$, the matrix function $w_{A}(x,\lambda)$ satisfies the system

[TABLE]

where the coefficient $\widetilde{q}_{0}(x)$ is given by the formula

[TABLE]

Remark 2.3

Formulas (2.2) and (2.5) yield

[TABLE]

and so $S(x)>0$ for $x\geq 0$ under additional condition $S(0)>0$. In particular, the condition of invertibility of $S(x)$ from Theorem 2.2 is fulfilled automatically when ${\mathcal{I}}=[0,\,\infty)$ and $S(0)>0$.

The matrix functions $S(x)^{-1}$, $X(x)$ and $w_{A}(x,\lambda)$ are well-defined in this case.

According to Theorem 2.2, the multiplication by $w_{A}$ transforms each solution $w$ of (2.1) into the solution $\widetilde{w}=w_{A}w$ of the system $\widetilde{w}^{\prime}=\widetilde{G}\widetilde{w}$ with the coefficients of $\widetilde{G}$ given by (2.10) and (2.11). This transformation of the solutions $w$ and coefficients $q_{k}$ is called GBDT. Matrix function $w_{A}$ is the so called Darboux matrix. The right hand side of (2.8) (with the additional property (2.7) and $x$ fixed) has the form of the Lev Sakhnovich transfer matrix function [30, 31, 33].

Under the conditions of Theorem 2.2 we have also

[TABLE]

Clearly, the definition (2.11) of $X$ and formula (2.6) imply that

[TABLE]

3 Explicit solutions of the dynamical system (1.1) and GBDT of the matrix Schrödinger equation

1.

Let us write down the coefficient $\widetilde{q}_{0}$ of the transformed system in the block form. We partition $\Pi$ into two $h\times h$ blocks and partition $X$ introduced in (2.11) into four $h\times h$ blocks:

[TABLE]

Thus, $\widetilde{q}_{0}$ in Theorem 2.2 (see (2.11)) has the form

[TABLE]

In order to rewrite (2.13) in a more convenient form, we shall need also the block representation of $\Pi^{*}S^{-1}$, $q_{1}^{*}$ and $\widetilde{q}_{0}^{*}$:

[TABLE]

which follows from (2.2), (2.14) and (3.2). Now, (2.13) takes the form

[TABLE]

Differentiating the second equality in (3.7) (and taking into account the first equality), we obtain

[TABLE]

Using (3.8) we derive the main theorem in this section

Theorem 3.1

Let the parameter matrices $A$, $S(0)=S(0)^{*}$ and $\Pi(0)$ be chosen so that (2.4) is valid, let the $h\times h$ potential $u=u^{*}$ be locally summable on ${\mathbb{R}}$, and introduce $\Pi(x)$ and $S(x)$ via (2.5) where (2.2) holds.

Then, in the points of invertibility of $S(x)$, the matrix function

[TABLE]

satisfies the continuous dynamical Schrödinger system

[TABLE]

where $\widetilde{u}=\widetilde{u}^{*}$ is given by the formula

[TABLE]

P r o o f

. Taking into account (2.5), (2.13) and (2.2), (3.6), we calculate $X_{22}^{\prime}$:

[TABLE]

In view of (3.12) (and definition (3.11) of $\widetilde{u}$), we rewrite (3.8) in the form

[TABLE]

According to (3.5), we have $z_{2}=\begin{bmatrix}0&I_{h}\end{bmatrix}\Pi^{*}S^{-1}$. Therefore, (3.9) and (3.13) imply (3.10). We also note that $\widetilde{u}=\widetilde{u}^{*}$ is immediate from $u=u^{*}$ and formulas (2.14) and (3.11).

$\blackbox$

Remark 3.2

Under conditions $S(0)>0$ and $x\in[0,\,\infty)$, the matrix function $S(x)$ is invertible $($recall Remark 2.3$)$. Thus, the matrix functions $S(x)^{-1}$, $X(x)$, $\widetilde{u}(x)$ and $\psi(x)$ $($considered in Theorem 3.1$)$ are well-defined under these conditions.

2.

If the conditions of Theorem 3.1 and Remark 3.2 are valid, we obtain the following corollary.

Corollary 3.3

Consider dynamical Schrödinger equations on ${\mathcal{I}}=[0,\infty)$, let the conditions of Theorem 3.1 hold, and assume that $S(0)>0$. Then, the columns of the matrix function $\begin{bmatrix}0&I_{h}\end{bmatrix}\Pi(x)^{*}S(x)^{-1}$ belong to $L^{2}_{h}(0,\infty)$ $($i.e., these columns are squarely summable$)$ and the solutions $\psi(x,t)g$ $(g\in{\mathbb{C}}^{h})$ of system (3.10) belong to $L^{2}_{h}(0,\infty)$ for each fixed $t$.

P r o o f

. In view of the second equality in (2.5) and the first equality in (2.12), we have

[TABLE]

Formula (3.14) implies that

[TABLE]

which proves the corollary.

$\blackbox$

Remark 3.4

For the study of the dependence on time of the solutions (3.6) and (4.24) of the discrete and continuous, respectively, Schrödinger equations, one may use the representation of $A$ in Jordan normal form:

[TABLE]

The Jordan representation above yields the equality

[TABLE]

Taking into account formula (3.9), Corollary 3.3 and representation (3.17), we see that the following asymptotics is valid generically$:$

[TABLE]

*where $\|\cdot\|$ is the norm in $L^{2}_{h}(0,\infty)$, $g\in{\mathbb{C}}^{h}$, $\tau_{+}=\max_{1\leq i\leq N}\Im(\lambda_{i})$,

$\tau_{-}=\min_{1\leq i\leq N}\Im(\lambda_{i})$, $\,\displaystyle{r_{\pm}=\max_{\Im(\lambda_{i})=\tau_{\pm}}(n_{i}-1)}$.*

We note that in a different way the Jordan structure of $A$ was used in [25] to study (and explain) an interesting multi-lump phenomena discovered in [1].

3.

Using considerations similar to those in Paragraph 1 of this section, we construct GBDT for matrix Schrödinger equation (1.4). Solution $w$ of system (2.1) with the coefficients given by (2.2) can be written down in the block form: w=\left[\begin{array}[]{c}y\\ \hat{y}\end{array}\right] ($y,\hat{y}\,\in{\mathbb{C}}^{h}$). Hence, we rewrite (2.1) as

[TABLE]

that is, (1.4) is fulfilled. So system (2.1), (2.2) is equivalent to the Schrödinger equation (1.4). The following proposition is a corollary of Theorem 2.2.

Proposition 3.5

Let a vector function $y(x,\lambda)$ satisfy the Schrödinger equation (1.4), where the $h\times h$ potential $u=u^{*}$ is locally summable on ${\mathbb{R}}$, and assume that the conditions of Theorem 2.2 hold. Then, the vector function

[TABLE]

with $w=\begin{bmatrix}y\\ y^{\prime}\end{bmatrix}$, satisfies the matrix Schrödinger equation

[TABLE]

where $\widetilde{u}=\widetilde{u}^{*}$ is given by the formula (3.11).

P r o o f

. According to Theorem 2.2, we have $\widetilde{w}^{\prime}=\widetilde{G}\widetilde{w}$. We rewrite this equation in terms of the blocks $\widetilde{y}$ and $\breve{y}:=[0\quad I_{h}]\widetilde{w}$ of $\widetilde{w}$:

[TABLE]

Differentiating $\widetilde{y}$ in the first equation above and using the second equation, we obtain $\widetilde{y}^{\prime\prime}=-\lambda\widetilde{y}+(u+X_{12}+X_{21}+X_{22}^{2}-X_{22}^{\prime})\widetilde{y}.$ Now, using (3.12), we derive

[TABLE]

Relation (3.19) is immediate from (3.11) and (3.20).

$\blackbox$

Instead of the Schrödinger equation (3.19) one can talk about Schrödinger operator ${\widetilde{\mathcal{L}}}=-\frac{d}{dx^{2}}+{\widetilde{u}}$ with a properly defined domain.

4.

If $\Pi$ and $S$ are known explicitly, then representations (3.9) and (3.18) provide explicit solutions of Schrödinger systems (3.10) and (3.19), respectively. In particular, $\Pi$ and $S$ are easily constructed in the case $u\equiv 0$ (see [13]). For this purpose we partition $\Pi$ into $n\times h$ blocks: $\Pi=\begin{bmatrix}\Lambda_{1}&\Lambda_{2}\end{bmatrix}$. Then, the first equation in (2.5) takes (for $u\equiv 0$) the form

[TABLE]

Remark 3.6

When $u\equiv 0$, then $\Pi(x)$ in Theorems 2.2 and 3.1 and in Proposition 3.5 is given by the formulas $\Pi(x)=\begin{bmatrix}\Lambda_{1}(x)&\Lambda_{2}(x)\end{bmatrix}$ and

[TABLE]

which is immediate from (3.21). According to (2.5), the matrix function $S(x)$ is given by the formula

[TABLE]

Recall that we know $($that is, we choose$)$ parameter matrices $A$, $S(0)$ and $\Pi(0)$ or, equivalently, $A$, $S(0)$ and $\Lambda_{k}(0)$ $(k=1,2)$ which determine GBDT-transformation.

4 Discrete dynamical Schrödinger system

1.

GBDT (generalized Bäcklund-Darboux transformation) was applied to important linear and nonlinear discrete systems in [9, 10, 16, 26, 30]. In particular, discrete canonical systems and non-Abelian Toda lattices were studied in [26]. Jacobi matrices corresponding to explicit solutions of matrix Toda lattices were considered in [26, Appendix]. Using some modification of the results from [26, Appendix], we construct here explicit solutions of discrete dynamical Schrödinger systems. We present also direct proofs of the corresponding modified results from [26, Appendix], whereas in [26, Appendix] several essential facts are proved indirectly (via the theory of discrete canonical systems developed in the previous sections of [26]) and some details of the proofs are omitted.

We start with introducing generalized Bäcklund-Darboux transformation (GBDT) of block Jacobi matrices. Suppose that the sets of $h\times h$ matrices $\{C(k)\}_{k>0}$ and $\{Q(k)\}_{k>0}$ such that

[TABLE]

are given. The corresponding initial Jacobi matrix is introduced by the relations

[TABLE]

where $k>0$ and (according to (4.1) and (4.7)) $b_{k}=b_{k}^{*}$.

Recall that GBDT is determined by three parameter matrices. Thus, we fix $n>0$, two $n\times n$ parameter matrices $A$ and $S_{0}>0$ and an $n\times m$ $(m=2h)$ parameter matrix $\Pi_{0}$ such that

[TABLE]

Everywhere in this section $j$ is given by the second equality in (4.8). Introduce matrices $\Pi_{k}$ and $S_{k}$ for $k>0$ by the recursions

[TABLE]

where

[TABLE]

The following properties easily follow from (4.1) and (4.10): $jPj=I_{m}-P$,

[TABLE]

Therefore, taking adjoints of both parts of the first equality in (4.9) (and multiplying the result by $\mathrm{i}^{k}j$) we obtain an equivalent to this equality relation

[TABLE]

Remark 4.1

Setting in (4.12) $\mathrm{i}^{k}j\Pi_{k}^{*}=W(k)$ and $A^{*}=z$, we obtain an auxiliary linear system $($10.1.9$)$ from [32] for the matrix Toda chain, which explains the choice of the equation on $\Pi_{k}$ in (4.9). Namely, we see that this equation is a generalized auxiliary system for Toda chain with the generalized eigenvalue $A$.

Since $S_{0}>0$ and $C(k)>0$, the second equality in (4.9) yields $S_{k}>0$ for $k\geq 0$. Setting

[TABLE]

we define the transformed matrices $\widetilde{C}(k)$ and $\widetilde{Q}(k)$ via relations

[TABLE]

Clearly $X(k)\geq 0$ for $k\geq 0$, and so $\widetilde{C}(k)>0$ for $k>0$. Then, the transformation (GBDT) $\widetilde{\mathcal{J}}$ of the block Jacobi matrix ${\mathcal{J}}$ is defined by the equalities

[TABLE]

We note that formulas (4.20) and (4.21) coincide (after removal of tildes) with the formulas (4.6) and (4.7) which define ${\mathcal{J}}$.

According to [26, Appendix], we have $\widetilde{b}_{k}=\widetilde{b}_{k}^{*}$. Slightly modifying the proof of [26, Theorem A.1], one may derive that under condition

[TABLE]

we have

[TABLE]

Theorem 4.2

Suppose that Jacobi matrix $\widetilde{\mathcal{J}}$ is given by the formulas (4.20) and (4.21), that relations (4.1), (4.8) and (4.22) are valid, and that the matrices $\widetilde{C}(k)$ and $\widetilde{Q}(k)$ in (4.21) are given by (4.9)–(4.15).

Then the block vector function

[TABLE]

where $Y$ is introduced in (4.23), satisfies the discrete dynamical Schrödinger system $\mathrm{i}\left(\frac{\partial}{\partial t}\Psi\right)(t)=\widetilde{\mathcal{J}}\Psi(t)$.

Theorem 4.2 is immediate from (4.23) and it remains to prove (4.23). More precisely, we prove the following theorem.

Theorem 4.3

Suppose that the relations (4.1) and (4.8) $($where $S_{0}>0)$ are valid. Then the matrices $\widetilde{C}(k)$ and $\widetilde{Q}(k)$ given by (4.9)–(4.15) are well-defined and satisfy relations

[TABLE]

We also have $\widetilde{b}_{k}=\widetilde{b}_{k}^{*}$ for the matrices $\widetilde{b}_{k}$ in (4.21).

If, in addition, (4.22) holds, then the matrix $\widetilde{\mathcal{J}}$ of the form (4.20), (4.21) satisfies (4.23).

P r o o f

. Step 1. Taking into account the inequalities $S_{0}>0$, $C(k)>0$ and relations (4.9), (4.14), we explained already that $S_{k}>0$ $(k\geq 0)$ and that $\widetilde{C}(k)>0$ $(k>0)$. Therefore, the matrices $\widetilde{Q}(k)$ and $\widetilde{C}(k)$ are well-defined, the inequality for $\widetilde{C}(k)$ in (4.25) is valid, $\widetilde{C}(k)$ is invertible, and $\widetilde{\mathcal{J}}$ is also well-defined.

Using (4.8) we show by induction that the matrix identity

[TABLE]

holds for all $k\geq 0$. Namely, let us assume that the identity

[TABLE]

is valid for some $k>0$. Then, in view of the second equality in (4.9) we have

[TABLE]

On the other hand, the first equality in (4.9) and relations (4.11) imply that

[TABLE]

(Here we used also the equality $\xi(k)^{-1}=j\xi(k)^{*}j$, which is immediate from (4.11).) Comparing (4.27) and (4.28) we obtain (4.26).

Step 2. Next, we prove the equality

[TABLE]

Indeed, taking into account the second relation in (4.9), the equality

[TABLE]

which is immediate from (4.11), and the equalities

[TABLE]

we derive

[TABLE]

In view of the first relation in (4.9) and the equality $j\xi(k)^{*}j=\xi(k)^{-1}$, we rewrite (4.31) in the form

[TABLE]

Multiplying both sides of (4.32) by $\Pi_{k}^{*}$ from the left and using again the first relation in (4.9), we see that

[TABLE]

Substituting (into (4.26)) $k-1$ instead of $k$, we rewrite the result in the form

[TABLE]

After substitution of (4.34) into (4.33), we obtain

[TABLE]

Equality (4.29) follows from (4.35).

Step 3. Recall that $\xi(k)$ is $j$-unitary, that is, the relation $\xi(k)j\xi(k)^{*}=j$ (or, equivalently, $\xi(k)^{*}j\xi(k)=j$) in (4.11) holds. It is important to show that the transformed matrix $\widetilde{\xi}(k)$ introduced in (4.30) is $j$-unitary as well. Taking into account (4.14) and (4.15) we rewrite (4.30) in the form

[TABLE]

Assuming that $\det A\not=0$, we prove the equality

[TABLE]

We note that $\breve{w}(k)=w_{A}(k,0)$, where $w_{A}(k,\lambda)=I_{m}-\mathrm{i}j\Pi_{k}^{*}S_{k}^{-1}(A-\lambda I_{n})^{-1}\Pi_{k}$ (with matrices $A,\Pi_{k},S_{k}$ satisfying (4.26)) is the transfer matrix function in Lev Sakhnovich form. (Compare with $w_{A}$ in (2.8).) According to [31] (see also [30, p. 24]) we have $w_{A}(k,\lambda)jw_{A}(k,\overline{\lambda})^{*}=j$, and so the matrices $\breve{w}(k)$ ($k\geq 0$) are $j$-unitary. Thus, (4.37) implies that $\widetilde{\xi}(k)$ is $j$-unitary. If $\det A=0$, we have $\det(A-\lambda I_{n})\not=0$ for smal values $\lambda=\overline{\lambda}$ and approximate $\widetilde{\xi}(k)$ with the $j$-unitary matrices $\xi_{\lambda}(k)$ corresponding to the GBDT-generating triples $A-\lambda I_{n},\,S_{0},\,\Pi_{0}$. Hence, the equality (4.37) for the case $\det A\not=0$ yields the $j$-unitarity property

[TABLE]

without restriction on $\det A$. It remains to show that (4.37) is valid.

Indeed, let us rewrite (4.37) in the form

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Using the first relation in (4.9), we rewrite (4.39) in another equivalent form

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Substituting the expression for $\Pi_{k}^{*}S_{k}^{-1}$ from (4.29) into the second right hand term in (4.40), we obtain

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Therefore, using (4.36) and canceling similar terms we derive

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Recalling that by definition $X(k-1)=\Pi_{k-1}^{*}S_{k-1}^{-1}\Pi_{k-1}$ we further simplify the equality above, and so the following relation:

[TABLE]

is equivalent to (4.37). In view of (4.9) and (4.10), we have

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which proves (4.42). Thus, (4.37) is proved as well, and hence (4.38) holds.

In particular, (4.38) yields

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According to (4.30), formula (4.43) is equivalent to the equalities

[TABLE]

Therefore, the first equality in (4.25) is valid, and we also rewrite (4.30) in the form

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Comparing (4.45) and the first equality in (4.10), we see that the representations of $\widetilde{\xi}(k)$ and $\xi(k)$ differ only by tildes in the notations.

The equality $\widetilde{b}_{k}=\widetilde{b}_{k}^{*}$ (for $\widetilde{b}_{k}$ given by (4.21)) is immediate from the first relation in (4.25).

Step 4. Finally, let us prove (4.23). Using equalities (4.21) and the definition of $Y$ in (4.23), we obtain

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Relations (4.29), (4.30) and (4.46), (4.47) imply that

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In particular, taking into account (4.22), we derive

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From (4.29) and (4.45) we see that

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According to (4.21) and (4.50) we have

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Now, formulas (4.48) and (4.51) yield for $k>1$ that

[TABLE]

Equalities (4.49) and (4.52) imply (4.23).

$\blackbox$

Acknowledgments. The research of A.L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant No. P29177.