Weyl Titchmarsh Research Articles

Inverse spectral Love problem via Weyl–Titchmarsh function

In this paper, we prove an inverse resonance theorem for the half-solid with vanishing stresses on the surface via Weyl–Titchmarsh function. Using a semi-classical approach, it is possible to simplify this three-dimensional problem of the elastic wave equation for the half-solid as a Schrödinger equation with Robin boundary conditions on the half-line. The goal of the paper is to establish a method to recover the potential from eigenvalues and resonances through the Weyl–Titchmarsh function for non-self-adjoint problems and to establish a one-to-one and onto map between suitable function spaces. Moreover, we produce an algorithm in order to retrieve the shear modulus from the eigenvalues and resonances, via the spectral data.

The Jacobi Operator on (-1,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-1,1)$$\\end{document} and Its Various m-Functions

We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression τα,β=-(1-x)-α(1+x)-β(d/dx)((1-x)α+1(1+x)β+1)(d/dx),α,β∈R,x∈(-1,1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ au _{\\alpha ,\\beta } =&- (1-x)^{-\\alpha } (1+x)^{-\\beta }(d/dx) \\big ((1-x)^{\\alpha +1}(1+x)^{\\beta +1}\\big ) (d/dx), \\\\&\\alpha , \\beta \\in {\\mathbb {R}}, \\, x \\in (-1,1), \\end{aligned}$$\\end{document}in L2((-1,1);(1-x)α(1+x)βdx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2\\big ((-1,1); (1-x)^{\\alpha } (1+x)^{\\beta } dx\\big )$$\\end{document}, α,β∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha , \\beta \\in {\\mathbb {R}}$$\\end{document}. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.

The local Borg–Marchenko uniqueness theorem of matrix‐valued Dirac‐type operators for coefficients locally smooth at the right endpoint

AbstractWe provide an expression of the Weyl–Titchmarsh matrix associated with a self‐adjoint matrix‐valued Dirac‐type operator defined on [0, b) for . As a concrete application of it, we establish asymptotics of the difference of two Weyl–Titchmarsh matrices corresponding to two Dirac‐type operators , with a.e. on [0, a], in with , when the are sufficiently smooth in a right neighborhood of the point a and their right derivatives at a coincide up to a certain order. In addition to this, we also provide new proofs of the local Borg–Marchenko theorem and asymptotic high‐energy expansion of Weyl–Titchmarsh matrices associated with Dirac‐type operators.

Resolvent and spectrum for discrete symplectic systems in the limit point case

The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl–Titchmarsh M+(λ)-function. Furthermore, a dependence of the spectrum on a boundary condition is investigated and, consequently, several results of the singular Sturmian theory are derived.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

Discrete Self-adjoint Dirac Systems: Asymptotic Relations, Weyl Functions and Toeplitz Matrices

We consider discrete Dirac systems as an approach to the study of the corresponding block Toeplitz matrices, which in many ways completes the famous approach via Szegő recurrences and matrix orthogonal polynomials. We prove an analog of the Christoffel–Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl–Titchmarsh functions of discrete Dirac systems). These asymptotic relations are expressed also in terms of block Toeplitz matrices. We study the case of rational Weyl–Titchmarsh functions (and GBDT version of the Backlund–Darboux transformation of the trivial discrete Dirac system) as well. It is shown that block diagonal plus block semi-separable Toeplitz matrices (which are easily inverted) appear in this case.

Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter

Eigenfunctions expansion for discrete symplectic systems on a finite discrete interval is established in the case of a general linear dependence on the spectral parameter as a significant generalization of the Expansion theorem given by Bohner et al. (2009) [14]. Subsequently, an integral representation of the Weyl–Titchmarsh M(λ)-function is derived explicitly by using a suitable spectral function and a possible extension to the half-line case is discussed. The main results are illustrated by several examples.

On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

We study realizations generated by the original Weyl–Titchmarsh functions $$m_\infty (z)$$ and $$m_\alpha (z)$$ . It is shown that the Herglotz–Nevanlinna functions $$(-\,m_\infty (z))$$ and $$(1/m_\infty (z))$$ can be realized as the impedance functions of the corresponding Shrodinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions $$(-\,m_\alpha (z))$$ and $$(1/m_\alpha (z))$$ . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

On the classes of explicit solutions of Dirac, dynamical Dirac and Dirac–Weyl systems with non-vanishing at infinity potentials, their properties and applications

We construct explicitly potentials, Darboux matrix functions and corresponding solutions of Dirac, dynamical Dirac and Dirac–Weyl systems using generalised Bäcklund-Darboux transformation (GBDT) in the important case of nontrivial initial systems. In this way, we construct explicit solutions of systems with non-vanishing at infinity potentials, including steplike and power-law growth potentials. Thus, the constructed potentials (systems) differ fundamentally from the actively studied case of GBDT for the trivial initial systems.Generalised matrix eigenvalues A (not necessarily diagonal) and corresponding generalised matrix eigenfunctions Π(x) of the nontrivial initial systems are used in the GBDT constructions in this paper. Explicit expressions for these Π(x) are new and the method of deriving these expressions may be applied to various other important problems.The case of Dirac–Weyl system, which is of interest in electron dynamics and graphene theory is studied in greater detail, and generalised separation of variables appears in our approach to the study of this system.Explicit expressions for Weyl–Titchmarsh functions (in the form of pseudo-realisations) are derived.

On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below

We extend the classical boundary values(0.1)g(a)=−W(ua(λ0,⋅),g)(a)=limx↓a⁡g(x)uˆa(λ0,x),g[1](a)=(pg′)(a)=W(uˆa(λ0,⋅),g)(a)=limx↓a⁡g(x)−g(a)uˆa(λ0,x)ua(λ0,x), for regular Sturm–Liouville operators associated with differential expressions of the type τ=r(x)−1[−(d/dx)p(x)(d/dx)+q(x)] for a.e. x∈(a,b)⊂R, to the case where τ is singular on (a,b)⊆R and the associated minimal operator Tmin is bounded from below. Here ua(λ0,⋅) and uˆa(λ0,⋅) denote suitably normalized principal and nonprincipal solutions of τu=λ0u for appropriate λ0∈R, respectively.Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of τu=λ0u is closely related to a seminal 1992 paper by Niessen and Zettl [58]. We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of Tmin in the singular case in a manner familiar from the regular case.We briefly discuss the singular Weyl–Titchmarsh–Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.

The anisotropic Calderón problem for singular metrics of warped product type: the borderline between uniqueness and invisibility

In this paper, we investigate the anisotropic Calderón problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics ofwarped product type, that iswhose coefficients only depend on the horizontal direction of the cylinder. By singular, we mean that these coefficients are positive almost everywhere and belong to some L^p, 1 \leq p \leq \infty spaces only. Using the recent developments on Weyl–Titchmarsh’s theory for singular Sturm–Liouville operators, we prove that the local Dirichlet to Neumann maps at each end are well defined and determine the metric uniquely if Eventually, we show (in the warped product case and for zero frequency) that these uniqueness results are sharp by giving simple counterexamples for a class of singular metrics whose coefficients do not belong to the critical L^p space. All these counterexamples lead in fact to a region of space that is invisible to boundary measurements.

On square integrable solutions of a fractional differential equation

In this paper we construct the Weyl–Titchmarsh theory for the fractional Sturm–Liouville equation. For this purpose we used the Caputo and Riemann–Liouville fractional operators having the order is between zero and one.

Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree

Abstract We consider inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as the matrix potentials on each edge. As inverse data we use the Weyl–Titchmarsh matrix function or the dynamic response operator.

Inverse Scattering at Fixed Energy on Three-Dimensional Asymptotically Hyperbolic Stäckel Manifolds

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic Stäckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the Helmholtz equation can be separated into a system of a radial ODE and two angular ODEs. We can thus decompose the full scattering operator into generalized harmonics and the resulting partial scattering matrices consist of a countable set of $2 \\times 2$ matrices whose coefficients are the so-called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl–Titchmarsh functions associated with the radial ODE. Using a novel multivariable version of the Complex Angular Momentum method, we show that the knowledge of the scattering operator at a fixed non-zero energy is enough to determine uniquely the metric of the three-dimensional Stäckel manifold up to natural obstructions.

Singular Hamiltonian system with several spectral parameters

In this paper, the Weyl–Titchmarsh theory has been constructed for the singular 2n-dimensional (even order) Hamiltonian system with several spectral parameters. In particular, we consider that the left end point of the interval is regular and the right end point of the interval is singular for the Hamiltonian system with several parameters. Using the nested circles approach, we prove that at least n-linearly independent solutions are squarly integrable with respect to some matrix functions.

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.

Dependence of eigenvalues on the boundary conditions of Sturm–Liouville problems with one singular endpoint

This paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self-adjoint Sturm–Liouville problems with one singular endpoint. Locally continuous dependence of eigenvalues on the boundary conditions is proved. Especially, in the limit point case, the continuous dependence of isolated eigenvalues is shown by the relationships between Weyl–Titchmarsh m(λ)-function and the spectrum of the singular problems. Then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed. It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in Lw2 norm. Its proof is different from that in the regular Sturm–Liouville problems. The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.

Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points −n2 (n ∈ N) of (a physically appropriate generalization of) the Weyl– Titchmarsh m-function m(λ) for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis (‘Boundary-conditionvarying circle billiards and gratings: the Dirichlet singularity’, J. Phys. A: Math. Theor. 41 (2008) 135203). To achieve these results, we prove a limit-circle analogue to the limit-point m-function interpolation result of Rybkin and Tuan (‘A new interpolation formula for the Titchmarsh– Weyl m-function’, Proc. Amer. Math. Soc. 137 (2009) 4177–4185); however, our proof, using a Mittag-Leffler series representation of m(λ), involves a rather different method from theirs, circumventing the A-amplitude representation of Simon (‘A new approach to inverse spectral theory, I. Fundamental formalism’, Ann. Math. (2) 150 (1999) 1029–1057). Uniqueness of the potential then follows by appeal to a Borg–Marˇcenko argument.

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl–Titchmarsh M-functions and Donoghue-type M-functions corresponding to closed extensions of symmetric operators belong to it.The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl–Titchmarsh M-functions, respectively.

Density of Schrödinger Weyl–Titchmarsh m functions on Herglotz functions

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.