Hamiltonian systems and Sturm-Liouville equations: Darboux transformation and applications

We introduce GBDT version of Darboux transformation for symplectic and Hamiltonian systems as well as for Shin-Zettl systems and Sturm-Liouville equations. These are the first results on Darboux transformation for general-type Hamiltonian and for Shin-Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl-Titchmarsh functions and to the construction of explicit solutions of dynamical symplectic systems, of two-way diffusion equations and of indefinite Sturm-Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT.


Introduction
This paper is dedicated to the study of the important subclasses of the first order differential systems with a spectral parameter λ. Namely, we consider and so called Shin-Zettl systems Here J * is the conjugate transpose of the matrix J. We assume that the m × m (m ∈ N) matrix functions H 1 (x) and H 0 (x) in (1.1) and the functions p −1 , q, r 1 , r 2 and ω in (1.3) are locally summable on [0, ℓ) (ℓ ≤ ∞). The matrix function F in (1.3) is the 2 × 2 Shin-Zettl matrix of general form (see, e.g., § 2 in [13] or in [14]). We note that Shin-Zettl differential expressions were introduced in [47,50] and were actively studied in regularization and spectral theories (see the books [1,51], papers [13,14], recent surveys [35,52] and various references therein). The Lagrange-symmetric case ω = ω, p = p, q = q, r 1 = −r 2 (1.4) and the Lagrange-J-symmetric case r 1 = −r 2 (1.5) are of special interest [14]. Here µ stands for the value which is complex conjugate to µ. The entries of the 2 × 1 vector function y in (1.3) are denoted by y 1 and y 2 . When r 1 ≡ r 2 ≡ 0, we rewrite (1.3) in the form which is equivalent to the Sturm-Liouville equation where u = y 1 . If ω = ω, p = p and ω or p change signs, one speaks about indefinite Sturm-Liouville problem. Quasi-derivatives related to the quasi-derivatives generated by Shin-Zettl systems are used in the study of important modifications of Schrödinger-type operators (see, e.g., [12,46] and references therein) including Schrödinger-type operators with distributional potentials [12].
On the other hand, Lagrange-symmetric Shin-Zettl systems, where ω ≥ 0, form also a subclass of Hamiltonian systems. See, for instance, [21] on the representation (1.1), (1.2) of Hamiltonian systems and the equivalence of the definite Sturm-Liouville equation to a certain subclass of Hamiltonian systems. We note that the book [2] by Atkinson, the papers by Hinton and Shaw as well the Kac-Krein supplement [23] (to the translation of [2]) presented seminal developments in the theory of Hamiltonian systems and Sturm-Liouville equations. (For recent references on Hamiltonian systems see, e.g., [24,36,43,48].) In some works, conditions (3.1) are added in the definition of Hamiltonian systems but these conditions are absent in [21] and they are not essential for Darboux transformations, which we will construct here, as well.
In this paper we construct our GBDT version of the Bäcklund-Darboux transformation (see the results and references in [39,41,43]) for the cases of Hamiltonian and Shin-Zettl systems in order to study perturbations of these systems and corresponding transformations of the Weyl-Titchmarsh functions. We construct explicit solutions of the perturbed systems as well. Several versions of Bäcklund-Darboux transformations (see, e.g., [8,20,33,43] and references therein) are a well-known tool for the construction of explicit solutions of linear and integrable nonlinear equations. GBDT as well as Crum-Krein and commutation methods (which are related to Bäcklund-Darboux transformations) are also essential in the study of Weyl-Titchmarsh theory and important spectral problems [10,11,16,17,19,27,30,34,42].
As far as we know, neither Bäcklund-Darboux transformations nor commutation methods were applied to general-type Hamiltonian systems (1.1) and to Shin-Zettl systems (1.3) before (although commutation and Bäcklund-Darboux transformations for such important particular cases as Schrödinger equations, canonical systems and related Dirac equations are well-known).
We mention an interesting paper [5] on Kummer-Liouville transformation for Shin-Zettl systems but that transformation is different and was applied with different purposes.
Darboux transformation for symplectic and general-type Hamiltonian systems is introduced in Section 2. The corresponding transformations of the Weyl-Titchmarsh functions are considered in Section 3. GBDT for Shin-Zettl systems and Sturm-Liouville equations is introduced in Sections 4-6. Explicit solutions of dynamical symplectic systems and of two-way diffusion equations are constructed in Section 7. Finally, explicit solutions of indefinite Sturm-Liouville equations are considered in Section 8.
As usual, N denotes the set of natural numbers, C denotes the complex plane, C + is the open upper half-plane {λ : ℑ(λ) > 0} and C − is the open lower half-plane {λ : ℑ(λ) < 0}. The notation I n stands for the n × n identity matrix, H * is the conjugate transpose of the matrix H, the inequality H ≥ 0 means that H = H * and that all the eigenvalues of the matrix H are nonnegative.
When we deal with S(x) −1 , our further statements are valid in the points of invertibility of S(x). The questions of invertibility of S(x) are discussed in our sections separately (see, e.g., Remarks 2.3 and 8.1).
According to the subcase r = 1, l = 0 of [43,Theor. 7.4], the so called Darboux matrix for system (2.1) is given by the formula (2.5) More precisely, [43,Theor. 7.4] yields that w A satisfies the following equation We note that (in view of (2.4)) the matrix function w A (λ) of the form (2.5) is (for each x) the so called transfer matrix function in Lev Sakhnovich form (see [43][44][45] and references therein). System y ′ = F y is called the transformed (GBDT-transformed) system (recall that (2.1) is the initial system). An important step in the proof of (2.6) is the proof of the equation satisfies, in the points of invertibility of S(x), another (transformed) first order system where F (x, λ) is given by (2.7)-(2.9).

2.
The most important subcase of the considered above GBDT-transformations is the subcase of the initial system (2.1) such that (2.14) In that subcase we deal with system (1.1), where all the conditions (1.2) on Hamiltonian system, excluding the nonnegativity condition H 1 (x) ≥ 0, hold. If J * = J −1 (e.g., J has the form (3.1)) conditions (2.14) mean that system (1.1) is symplectic. Further in the paragraphs 2 and 3 we assume that the equalities (2.13) and (2.14) are valid. We omit indices in A 1 and Π 1 and set Using (2.13)-(2.15) we rewrite the first and second equations in (2.3), correspondingly, in the forms Thus, the equations on −ΠJ and on Π 2 coincide, and, in view of In this way, equations (2.3) are reduced to the equations Now, the matrix identity (2.4) and Darboux matrix (2.5) are rewritten in the form Moreover, using (2.13) and the equalities Formulas (2.14), (2.20) and (2.21) imply that H 0 = H * 0 , that is, F has the same form as F . Hence, the next proposition follows from Theorem 2.1.
3. If in the system (1.1) we have J = −J * = −J −1 and H 0 ≡ 0, we come to the important class of canonical systems. See GBDT for canonical system and its applications to Weyl-Titchmarsh theory in [40]. For the case of Hamiltonian systems with invertible J we can (similar to the case of canonical systems) consider transformation slightly different from (2.20), (2.21). More precisely, we introduce matrix functions w(x) and v(x, λ) by the formulas It is easy to see that w(x)J w(x) * = J, and so In view of Proposition 2.2 and relations (2.23) and (2.24), if y(x, λ) satisfies (1.1), then the matrix function y(x, λ) = v(x, λ)y(x, λ) satisfies the system where In the special case H 0 = icJ −1 (c = c), the formula (2.27) is simplified and we obtain H 0 ≡ icJ −1 .

Darboux transformations of Weyl-Titchmarsh functions
In his important paper [28], Krall introduced Weyl-Titchmarsh (or simply Weyl) M(λ)-functions of Hamiltonian systems in the classical terms of "Weyl circle" inequalities. Here, Weyl circles of system (1.1) on the intervals [0, ℓ ′ ] (ℓ ′ < ℓ) and the values λ in the upper half-plane λ ∈ C + (i.e., ℑ(λ) > 0) are considered. The Weyl circles in the lower half-plane C − are treated in a quite similar way and we omit that case. Krall required that m is even and that J in (1.1) has a special form: In fact, Hamiltonian system in [28] is written in a slightly different from (1.1) way and our J * stands for J in an equivalent to (1.1) system in [28].
Rewriting correspondingly the inequality for the Weyl circle (of matrices Here Y (x, λ) is the fundamental m × m solution of the Hamiltonian system (1.1) (such that (1.2) and (3.1) are valid), normalized by the initial condition According to Proposition 2.2, the fundamental solution Y (x, λ) (normalized by Y (0, λ) = E) of the transformed Hamiltonian system (2.22) is given by the formula Let us set where U ij (λ) are r × r blocks of U. In view of (3.4) and (3.5), the Weyl circle (of matrices M(λ)) for the transformed system on [0, ℓ ′ ] and for λ ∈ C + is determined by the inequality In this section we consider Hamiltonian systems and assume that S(0) > 0. Hence, according to Remark 2.3 we have S(x) > 0. Now, it is immediate from (3.7) that Using (3.9), we derive the next theorem. holds. Assume that M(λ) (λ ∈ C + ) belongs to the Weyl circle (3.2) of the system (1.1) and that where U is defined in (3.5). Then belongs to the Weyl circle of the transformed system.
P r o o f. Taking into account (3.11) and (3.12), we obtain Now, substitute (3.13) into the right-hand side of (3.9) and use (3.2) in order to see that (3.6) is valid.
In the limit point case (see, e.g., the discussions in [22,29]) there is a unique holomorphic in C + Weyl function M(λ) the values of which belong to all the Weyl circles (3.2) such that ℓ ′ < ℓ (λ ∈ C + ). We note that M(λ) is the limit of the values of M(λ) when ℓ ′ tends to ℓ. Thus, formula (3.12) shows that is a Weyl function of the transformed system considered on [0, ℓ).

GBDT for Shin-Zettl systems
Shin-Zettl systems (1.3) present (as well as Hamiltonian systems) an important subclass of systems (2.1). Matrices Q 1 and Q 2 , in the case of Shin-Zettl systems, have the form Recall that GBDT is determined by the parameter matrices A 1 , A 2 , S(0), Π 1 (0) and Π 2 (0) such that (2.2) holds. For the Shin-Zettl systems, we have m = 2, and so matrices Π 1 (0) and Π 2 (0) are n × 2 matrices. Using the second equality in (1.3) and the first equality in (4.1), we rewrite F given by (2.7)-(2.9) in the Shin-Zettl form , ω = ω, p = p; (4.2) where X ik (x) are the entries of X(x). Now, the following proposition is immediate from Theorem 2.1.  In the next section, we consider the Lagrange-symmetric case (i.e., the case (1.4)).
We note that Q 1 and Q 0 given by (5.1) admit representation (2.13), where we see that the formulas of §2 in Section 2 are valid for Lagrange-symmetric case.

Sturm-Liouville equations
In this section we consider Sturm-Liouville equation (1.7). GBDT for its particular case (namely, for Schrödinger equation where p ≡ ω ≡ 1) was dealt with in [18] but the general equation (1.7) contains other interesting subcases, where GBDT could be useful as well.
Proposition 6.1 Let the function pω be differentiable and its derivative (pω) ′ as well as the functions p −1 , q and ω be locally summable on [0, ℓ). Assume that ω = ω, p = p, q = q, r ≡ 0, (6.1) and set where w A is given by the relations (5.9) and (5.8), H 0 and H 1 (in (5.8)) are given by (5.4) and y satisfies the initial Lagrange-symmetric Shin-Zettl equation Then the entry y 1 of y satisfies the transformed Sturm-Liouville equation 5) and X ik are the entries of X given by (5.6).
P r o o f. Recall that in Section 5 we rewrote Shin-Zettl system in the form (5.3) where u = y 1 . In the notations of the transformed system it means − p( y ′ 1 − r y 1 ) ′ − rp( y ′ 1 − r y 1 ) + q y 1 = λω y 1 , (6.6) where r := r 1 (x) = − r 2 (x). Using the identity r 1 ≡ r 2 ≡ 0 and equalities (4.3) and (5.7) we present (6.6) in the form which is equivalent to (6.4) with Finally, in order to show that the functionsq given by (6.5) and (6.8) coincide, let us differentiate X 12 . Taking into account (5.6) and (5.8), we obtain: In particular, for X 12 we obtain Here we again took into account that r 1 ≡ r 2 ≡ 0. Equalities (6.8) and (6.9) imply (6.5).  Formally applying Laplace transform to the system (1.1) (satisfying (2.14)), we come to the interesting dynamical system When J * = J −1 system (7.1) is a dynamical symplectic system. In order to construct Darboux transformation of system (7.1) and solutions of the transformed system, we use (2.17) and rewrite (2.10) (for our case where the relations (2.13)-(2.15) are valid) in the form 2) We note that (7.3) is equivalent to the second equality in (2.20).
When H 1 ≥ 0, the energy E z (t) of the solutions z of system (7.1) on [0, a] (0 < a < ℓ) is given by the formula
Recall that (7.18) is an equation of the form (7.12).
The singularity ofq(x) at x = 0 is of interest. Some particular cases (but in greater detail) were considered in [27,Section 5], and it was proved for those cases thatq(x) = O(x −2 ) when x tends to 0.