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

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.


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., [2-4, 7, 8, 17, 21, 27, 28, 35] and numerous references therein). Continuous dynamical Schrödinger system has the form: i ∂ ∂t ψ(x, t) = Hψ (x, t), H := − ∂ 2 ∂x 2 + u(x) (u = u * ), (1.1) where u is an h × h matrix function, h ∈ N, and 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 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: where J is a semi-infinite block Jacobi matrix and Ψ is a block vector Here, the blocks a k , b k and c k are h × h matrices and c k+1 = a * k (k ≥ 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 −y ′′ (x, λ) + u(x)y(x, λ) = λy(x, λ) (u = u * ), y ′ := d dx y. (1.4) 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.

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 ′ = G(x, λ)w, G(x, λ) = − r k=−r λ k q k (x)) was treated in [23,30] (see also some references therein). First order system where w takes values in C m (m := 2h) and m × m coefficients q 1 and q 2 have the form is equivalent to the matrix Schrödinger equation (1.4) with a self-adjoint h × 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 I, that is, we assume that x ∈ I. Without loss of generality we assume also that 0 ∈ I and speak later about parameter matrices S(0) and Π(0) instead of S(x 0 ) and Π(x 0 ) for some fixed x 0 ∈ I. The most interesting for us is the case of the semiaxis I = [0, ∞).
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: Thus the conditions of Proposition 1.4 from [23] are fulfilled, and we may use this proposition and some formulas from its proof. 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 ∈ N. Then, we fix n × n matrices A and S(0) = S(0) * , and an n × m (m = 2h) matrix Π(0) such that the following matrix identity holds: Suppose that such parameter matrices are fixed and that the potential u(x) is locally summable on R. Now, we can introduce matrix functions Π(x) and S(x) with the values Π(0) and S(0) at x = 0 as the solutions of the linear differential equations where q 1 and q 0 are given by (2.2), and so q 1 j = (q 1 j) * . Thus, in view of Notice that equations (2.5) are constructed in such a way that the identity follows (for all x ∈ 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) ≡ 0 we can define a matrix function where λ ∈ σ(A) (σ means spectrum).
where the coefficient q 0 (x) is given by the formula

12)
and so S(x) > 0 for x ≥ 0 under additional condition S(0) > 0. In particular, the condition of invertibility of S(x) from Theorem 2.2 is fulfilled automatically when I = [0, ∞) and S(0) > 0. The matrix functions S(x) −1 , X(x) and w A (x, λ) 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 w = w A w of the system w ′ = G w with the coefficients of 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 Clearly, the definition (2.11) of X and formula (2.6) imply that (2.14) 3 Explicit solutions of the dynamical system (1.1) and GBDT of the matrix Schrödinger equation 1. Let us write down the coefficient q 0 of the transformed system in the block form. We partition Π into two h × h blocks and partition X introduced in (2.11) into four h × h blocks: Thus, q 0 in Theorem 2.2 (see (2.11)) has the form In order to rewrite (2.13) in a more convenient form, we shall need also the block representation of Π * S −1 , q * 1 and q * 0 : which follows from (2.2), (2.14) and (3.2). Now, (2.13) takes the form Differentiating the second equality in (3.5) (and taking into account the first equality), we obtain Using (3.6) we derive the main theorem in this section Theorem 3.1 Let the parameter matrices A, S(0) = S(0) * and Π(0) be chosen so that (2.4) is valid, let the h×h potential u = u * be locally summable on R, and introduce Π(x) and S(x) via (2.5) where (2.2) holds. Then, in the points of invertibility of S(x), the matrix function satisfies the continuous dynamical Schrödinger system where u = u * is given by the formula In view of (3.10) (and definition (3.9) of u), we rewrite (3.6) in the form According to (3.3), we have z 2 = 0 I h Π * S −1 . Therefore, (3.7) and (3.11) imply (3.8). We also note that u = u * is immediate from u = u * and formulas (2.14) and (3.9).
, these columns are squarely summable) and the solutions ψ( P r o o f. In view of the second equality in (2.5) and the first equality in (2.12), we have which proves the corollary. (3.14) The Jordan representation above yields the equality Taking into account formula (3.7), Corollary 3.3 and representation (3.15), we see that the following asymptotics is valid generically: 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 = ŷ y (y,ŷ ∈ C h ). Hence, we rewrite (2.1) as with w = y y ′ , satisfies the matrix Schrödinger equation where u = u * is given by the formula (3.9).
P r o o f. According to Theorem 2.2, we have w ′ = G w. We rewrite this equation in terms of the blocks y andy := [0 I h ] w of w: y ′ = −X 22 y +y,y ′ = −λ y + (u + X 12 + X 21 ) y + X 22y .
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 × h matrices {C(k)} k>0 and {Q(k)} k>0 such that where k > 0 and (according to (4.1) and (4.3)) b k = b * k . Recall that GBDT is determined by three parameter matrices. Thus, we fix n > 0, two n×n parameter matrices A and S 0 > 0 and an n×m (m = 2h) parameter matrix Π 0 such that Everywhere in this section j is given by the second equality in (4.4). Introduce matrices Π k and S k for k > 0 by the recursions The following properties easily follow from (4.1) and (4.6): jP j = I m − P , Therefore, taking adjoints of both parts of the first equality in (4.5) (and multiplying the result by i k j) we obtain an equivalent to this equality relation Remark 4.1 Setting in (4.8) i k jΠ * 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 Π k in (4.5). 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 (4.5) yields S k > 0 for k ≥ 0. Setting we define the transformed matrices C(k) and Q(k) via relations Clearly X(k) ≥ 0 for k ≥ 0, and so C(k) > 0 for k > 0. Then, the transformation (GBDT) J of the block Jacobi matrix J is defined by the equalities We note that formulas (4.12) and (4.13) coincide (after removal of tildes) with the formulas (4.2) and (4.3) which define J . According to [26,Appendix], we have b k = b * k . Slightly modifying the proof of [26, Theorem A.1], one may derive that under condition we have Theorem 4.2 Suppose that Jacobi matrix J is given by the formulas (4.12) and (4.13), that relations (4.1), (4.4) and (4.14) are valid, and that the matrices C(k) and Q(k) in (4.13) are given by (4.5)-(4.11).
Then the block vector function
Theorem 4.2 is immediate from (4.15) and it remains to prove (4.15). More precisely, we prove the following theorem. We also have b k = b * k for the matrices b k in (4.13). If, in addition, (4.14) holds, then the matrix J of the form (4.12), (4.13) satisfies (4.15).
P r o o f. Step 1. Taking into account the inequalities S 0 > 0, C(k) > 0 and relations (4.5), (4.10), we explained already that S k > 0 (k ≥ 0) and that C(k) > 0 (k > 0). Therefore, the matrices Q(k) and C(k) are well-defined, the inequality for C(k) in (4.17) is valid, C(k) is invertible, and J is also well-defined.
Using (4.4) we show by induction that the matrix identity holds for all k ≥ 0. Namely, let us assume that the identity is valid for some k > 0. Then, in view of the second equality in (4.5) we have On the other hand, the first equality in (4.5) and relations (4.7) imply that (4.20) (Here we used also the equality ξ(k) −1 = jξ(k) * j, which is immediate from (4.7).) Comparing (4.19) and (4.20) we obtain (4.18).
Step 2. Next, we prove the equality Indeed, taking into account the second relation in (4.5), the equality jξ(k) * P = ζ(k), which is immediate from (4.7), and the equalities j(I m − P )j = P, P (I m − P ) = 0, we derive In view of the first relation in (4.5) and the equality jξ(k) * j = ξ(k) −1 , we rewrite (4.23) in the form Multiplying both sides of (4.24) by Π * k from the left and using again the first relation in (4.5), we see that Substituting (into (4.18)) k − 1 instead of k, we rewrite the result in the form After substitution of (4.26) into (4.25), we obtain Equality (4.21) follows from (4.27).
In particular, (4.30) yields (I m − P ) ξ(k)j ξ(k) * (I m − P ) = 0, P ξ(k)j ξ(k) * (I m − P ) = I h . Comparing (4.37) and the first equality in (4.6), we see that the representations of ξ(k) and ξ(k) differ only by tildes in the notations. The equality b k = b * k (for b k given by (4.13)) is immediate from the first relation in (4.17).