We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\bbR$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.
We prove local and global versions of Borg-Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velazquez) with matrix-valued Verblunsky coefficients. While our half-lattice results are formulated in terms of matrix-valued Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's matrices. We also develop the basics of Weyl-Titchmarsh theory for CMV operators with matrix-valued Verblunsky coefficients as this is of independent interest and an essential ingredient in proving the corresponding…
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and the associated Green's matrices for whole-line and half-line self-adjoint Hamiltonian finite difference systems with separated boundary conditions.
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line.
We consider discrete Schroedinger operator J with Wigner-von Neumann potential not belonging to l^2. We find asymptotics of orthonormal polynomials associated to J. We prove the Weyl-Titchmarsh type formula, which relates the spectral density of J to a coefficient in asymptotics of orthonormal polynomials.
Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for this operator is proven.
We show that the Herglotz functions that arise as Weyl-Titchmarsh $m$ functions of one-dimensional Schrödinger operators are dense in the space of all Herglotz functions with respect to uniform convergence on compact subsets of the upper half plane. This result is obtained as an application of de Branges theory of canonical systems.
We develop singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results.
Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schroedinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.
We prove a generalization of the well-known theorems by Borg and Hochstadt for periodic self-adjoint Schrödinger operators without a spectral gap, respectively, one gap in their spectrum, in the matrix-valued context. Our extension of the theorems of Borg and Hochstadt replaces the periodicity condition of the potential by the more general property of being reflectionless (the resulting potentials then automatically turn out to be periodic and we recover Després' matrix version of Borg's result). In addition, we assume the spectra to have uniform maximum multiplicity (a condition automatically…
We establish a mutual relationship between main analytic objects for the dissipative extension theory of a symmetric operator $\dot A$ with deficiency indices $(1,1)$. In particular, we introduce the Weyl-Titchmarsh function $\cM$ of a maximal dissipative extension $\hat A$ of the symmetric operator $\dot A$. Given a reference self-adjoint extension $A$ of $\dot A$, we introduce a von Neumann parameter $κ$, $|κ|<1$, characterizing the domain of the dissipative extension $\hat A$ against $\Dom (A)$ and show that the pair $(κ, \cM)$ is a complete unitary invariant of the triple $(\dot A, A, \hat…
We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schrödinger operators $H_j = -\f{d^2}{dx^2} + q_j$, j=1,2 in $L^2 ((0,R))$, $0<R\leq \infty$, are exponentially close, that is, $|m_1(z)- m_2(z)| \underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a})$, 0<a<R, then $q_1 = q_2$ a.e.~on $[0,a]$. The result applies to any boundary conditions at x=0 and x=R and should be considered a local version of the celebrated Borg-Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover…
We prove local and global versions of Borg-Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velazquez \cite{CMV03}). While our half-lattice results are formulated in terms of Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's functions.
In this note it is proved that the complete spectral data of selfadjoint Schrödinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from si…
We develop singular Weyl-Titchmarsh-Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg-Marchenko and Hochstadt-Liebermann type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.
We provide a detailed treatment of Weyl-Titchmarsh theory for half-lattice and full-lattice Cantero-Moral-Velazquez (CMV) operators and discuss their systems of orthonormal Laurent polynomials on the unit circle, spectral functions, variants of Weyl--Titchmarsh functions, and Green's functions. In particular, we discuss the corresponding spectral representations of half-lattice and full-lattice CMV operators.
We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl -Titchmarsh matrix-valued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extens…
In this paper we give simplified formulas for certain polynomials which arise in some new Titchmarsh-Weyl m-functions for the radial part of the separated Hydrogen atom on the half line and two independent programs for generating them using the symbolic manipulator Mathematica.
The matrix-valued Weyl-Titchmarsh functions $M(λ)$ of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of $M(λ)$) and the residues of $M(λ)$ is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, $N\times N$ Weyl-Titchmarsh functions) corresponding to $N\times N$ self-adjoint square-integrable matrix-valued potentials is given, if all $N$ eigenvalues of the averaged potential are distinct.
We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of orthogonal polynomials associated with the matrix.
We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.
We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a genera…
In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary diff…
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.
We develop Weyl-Titchmarsh theory for self-adjoint Schrödinger operators $H_α$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $τ=-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $α= α^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(α)u'(a)+\cos(α)u(a)=0, $$ with $u$ lying in the domain of the underl…
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg-Marchenko type uniqueness theorems are derived for both discrete and continuous non-self-adjoint systems too.
We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ τf = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setu…
Inverse spectral problems for Sturm-Liouville operators with nonlocal boundary conditions are studied. As the main spectral characteristics we introduce the so-called Weyl-type function and two spectra, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Sturm-Liouville operator. Two uniqueness theorems of inverse problems from the Weyl-type function and two spectra are presented and proved, respectively.
We study the question to what extent spectral information of a Schrödinger operator on a finite, compact metric graph subject to standard or $δ$-type matching conditions can be recovered from a corresponding Titchmarsh-Weyl function on the boundary of the graph. In contrast to the case of ordinary or partial differential operators, the knowledge of the Titchmarsh-Weyl function is in general not sufficient for recovering the complete spectrum of the operator (or the potentials on the edges). However, it is shown that those eigenvalues with sufficiently high (depending on the cyclomatic number o…
We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
We investigate the relation between the eigenvalues of the Laplacian with Kirchhoff vertex conditions on a finite metric graph and a corresponding Titchmarsh-Weyl function (a parameter-dependent Neumann-to-Dirichlet map). We give a complete description of all real resonances, including multiplicities, in terms of the edge lengths and the connectivity of the graph, and apply it to characterize all eigenvalues which are visible for the Titchmarsh-Weyl function.
We derive Fourier integral associated to the complex Martin function in the Denjoy domain of Widom type with the Direct Cauchy Theorem (DCT). As an application we study reflectionless Weyl-Titchmarsh functions in such domains, related to them canonical systems and transfer matrices. The DCT property appears to be crucial in many aspects of the underlying theory.
We consider the Schrödinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy.
We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-κ^2) =-κ- \int_0^a A(α) e^{-2ακ} dα+O(e^{-(2a -ε)κ}) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure ρ: A(α) = -2\int_{-\infty}^\infty λ^{-\frac12} \sin (2α\sqrtλ)\, dρ(λ) (since the integral is divergent, this f…
Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral representation of the Weyl function, and results on interpolation of Weyl functions are also derived.
Discrete Dirac type self-adjoint system is equivalent to the block Szegö recurrence. Representation of the fundamental solution is obtained, inverse problems on the interval and semi-axis are solved. A Borg-Marchenko type result is obtained too. Connections with the block Toeplitz matrices are treated.
The main objective of this paper is to identify principal solutions associated with Sturm-Liouville operators on arbitrary open intervals $(a,b) \subseteq \mathbb{R}$, as introduced by Leighton and Morse in the scalar context in 1936 and by Hartman in the matrix-valued situation in 1957, with Weyl-Titchmarsh solutions, as long as the underlying Sturm-Liouville differential expression is nonoscillatory (resp., disconjugate or bounded from below near an endpoint) and in the limit point case at the endpoint in question. In addition, we derive an explicit formula for Weyl-Titchmarsh functions in t…
Weyl theory for Dirac systems with rectangular matrix potentials is non-classical. The corresponding Weyl functions are rectangular matrix functions. Furthermore, they are non-expansive in the upper semi-plane. Inverse problems are treated for such Weyl functions, and some results are new even for the square Weyl functions. High energy asymptotics of Weyl functions and Borg-Marchenko type uniqueness results are derived too.
A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szegö recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.
We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.
We investigate the connection between singular Weyl-Titchmarsh-Kodaira theory and the double commutation method for one-dimensional Dirac operators. In particular, we compute the singular Weyl function of the commuted operator in terms of the data from the original operator. These results are then applied to radial Dirac operators in order to show that the singular Weyl function of such an operator belongs to a generalized Nevanlinna class $N_{κ_0}$ with $κ_0=\lfloor|κ| + \frac{1}{2}\rfloor$, where $κ\in \mathbb{R}$ is the corresponding angular momentum.
In this paper inverse problems for Dirac operator with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are provided, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Dirac operator.
We explore the connections between singular Weyl-Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schroedinger operators (also known as Bessel operators). We also investigate the connections with the generalized Baecklund-Darboux transformation.
A non-classical Weyl theory is developed for skew-self-adjoint Dirac systems with rectangular matrix potentials. The notion of the Weyl function is introduced and direct and inverse problems are solved. A Borg-Marchenko type uniqueness result and the evolution of the Weyl function for the corresponding focusing nonlinear Schrödinger equation are also derived.
Inverse problems for differential pencils with nonlocal conditions are investigated. Several uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Sturm-Liouville operator.
We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.
We investigate the connections between Weyl-Titchmarsh-Kodaira theory for one-dimensional Schrödinger operators and the theory of $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be $n$-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz-Nevanlinna class. As an application we show that perturbed Bessel operators are $n$-entire, improving the previously known condition…
