Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

The class of two-dimensional trace-normed canonical systems of differential equations on is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (–, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.     

1-D Schrödinger operators with local point interactions on a discrete set

Spectral properties of 1-D Schrödinger operators with local point interactions on a discrete set are well studied when d_∗:=infn,k∈N|xn−xk|>0. Our paper is devoted to the case d_∗=0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions.We show that the spectral properties of HX,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators HX,α to be self-adjoint, lower semibounded, and discrete in the case d_∗=0.The operators with δ^′-type interactions are investigated too. The obtained results demonstrate that in the case d_∗=0, as distinguished from the case d_∗>0, the spectral properties of the operators with δ- and δ^′-type interactions are substantially different.     

A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge in the upper half-plane in the case of L² coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szegö-Kolmogorov-Krein condition is satisfied. Thus, we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.     

Small perturbations of canonical systems

We consider a singular two-dimensional canonical systemJy=–zHy on [0, ) such that at Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.     

The Bitangential Inverse Input Impedance Problem for Canonical Systems,I: Weyl-Titchmarsh Classification,Existence and Uniqueness

The inverse input impedance problem is investigated in the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [ArD1]. The set of solutions for a problem with a given input impedance matrix (i.e., Weyl- Titchmarsh function) is parameterized by chains of associated pairs of entire inner p × p matrix valued functions. In our considerations the given data for the inverse bitangential input impedance problem is such a chain and an input impedance matrix, i.e., a p × p matrix valued function in the Carathéodory class. Existence and uniqueness theorems for the solution of this problem are obtained by consideration of a corresponding family of generalized bitangential Carathéodory interpolation problems. The connection between the inverse bitangential input scattering problem that was studied in [ArD4] and the bitangential input impedance problem is also exploited. The successive sections deal with: 1. The introduction, 2. Domains of linear fractional transformations, 3. Associated pairs of the first and second kind, 4. Matrix balls, 5. The classification of canonical systems via the limit ball, 6. The Weyl-Titchmarsh characterization of the input impedance, 7. Applications of interpolation to the bitangential inverse input impedance problem. Formulas for recovering the underlying canonical integral systems, examples and related results on the inverse bitangential spectral problem will be presented in subsequent publications.D. Z. Arov thanks the Weizmann Institute of Science for hospitality and support, partially as a Varon Visiting Professor and partially through the Minerva Foundation. H. Dym thanks Renee and Jay Weiss for endowing the chair which supports his research and the Minerva Foundation.     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

The bitangential inverse spectral problem for canonical systems

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p×p matrix valued function σ(μ) on and a normalized monotonic continuous chain of pairs , of entire inner p×p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ(μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.     

Sturm-Liouville systems with rational Weyl functions: Explicit formulas and applications

This paper solves explicitly the direct and inverse spectral problems for Sturm-Liouville systems with rational Weyl functions. As in the previous papers of the authors on canonical and pseudocanonical systems, the method employed in the present paper is based on state space techniques and uses the idea of realization from mathematical system theory. For a subclass of rational potentials a property of modified bispectrality is proven. A wide class of solutions of the matrix Korteweg-de Vries equation is derived.To the memory of M.G. Kreîn, one of the founding fathers of modern spectral theory of differential operators, with admiration and gratitude.     

Bound states of a canonical system with a pseudo-exponential potential

Explicit formulas are given for the bound states (theL ²-eigenfunctions) and the corresponding eigenvalues of a self-adjoint operator defined by a canonical system with a pseudo-exponential potential. The formulas are expressed in terms of three matrices determining the potential. Both the half line and the full line case are considered.     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations

In this paper we present a new approach to the spectral theory of non-uniformly continuous functions and a new framework for the Loomis-Arendt-Batty-Vu theory. Our approach is direct and free of C₀-semigroups, so the obtained results, that extend previous ones, can be applied to large classes of evolution equations and their solutions.     

The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

We explore the sparsity of Weyl–Titchmarsh m-functions of discrete Schrödinger operators. Due to this, the set of their m-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral theory for discrete Schrödinger operators via their spectral measures should be difficult. To obtain the result, de Branges theory of canonical systems is applied to work on them, instead of Weyl–Titchmarsh m-functions.     

A Prüfer angle approach to singular Sturm–Liouville problems with Molčanov potentials

Singular Sturm–Liouville problems for -y^″+qy=λy$-y^{″}+\mathit{qy}=\lambda y$ on (0,∞)$(0,\infty )$ are studied for potentials q which are bounded below and satisfy Mol?anov's necessary and sufficient condition for discrete spectrum. A Prüfer angle approach is given for eigenvalue location and eigenfunction oscillation, paralleling that for the regular case. In particular, the eigenvalues are characterized by a “right-hand boundary condition” even though q is of limit point type.     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

Spectral properties of the Laplacian on bond-percolation graphs

Bond-percolation graphs are random subgraphs of the d-dimensional integer lattice generated by a standard bond-percolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, self-adjoint, ergodic random operators with off-diagonal disorder. They possess almost surely the non-random spectrum [0, 4d] and a self-averaging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the non-percolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge,where the exponent equals 1/2.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.