Steve Clark, Fritz Gesztesy
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.
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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.
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.
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.
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 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…
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…
Asymptotics of Titchmarsh--Weyl functions near the real axis andan application to the KdV hierarchy
本文推导了一维Schrödinger与Dirac算子的Titchmarsh–Weyl函数在边界趋近谱的多项式速率区域内的高能渐近展开，验证了Kotani构造KdV流的高能假设，证明了对应势可生成KdV层级指定成员的全局经典解。
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.
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.
The boundary control approach to the Titchmarsh-Weyl $m-$function
本文将边界控制理论与Titchmarsh-Weyl理论相结合，为Simon的$A-$幅度提供自然解释，并提出一种计算Schrödinger算子在Dirichlet边界条件下Titchmarsh-Weyl $m-$函数的高效新方法。
Block Jacobi matrices and Titchmarsh-Weyl function
本文研究块约当矩阵的泛化概念，包括蒂奇马什-韦伊尔函数、非子协调性等，提出新方法并改进已有结果。
On a New Algorithm for the Computation of Enclosures for the Titchmarsh-Weyl m-Function
本文提出一种新算法，用于计算Titchmarsh-Weyl m函数的验证包络，并指出在特定情况下需结合数学分析补充Lohner的AWA算法。
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.
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…
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 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 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 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…
Weil-Titchmarsh theory as applied to the singular non-sectorial Schrödinger operator. Conditions for discreteness of the spectrum and compactness of the resolvent
针对取值区域超出半平面的复值势导致奇异 Schrödinger 算子非扇形且数值域覆盖整个复平面的问题，本文利用 Weil-Titchmarsh 理论给出谱离散及预解式紧的充分条件。
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…
Inverse spectral Love problem via Weyl-Titchmarsh function
本文利用 Weyl-Titchmarsh 函数证明了半实体逆共振定理，将三维弹性波方程简化为半直线上的 Schrödinger 方程，建立了非自伴问题中恢复势函数的方法，并提出从特征值和共振提取剪切模量的算法。
On a conjecture of Bennewitz, and the behaviour of the Titchmarsh-Weyl matrix near a pole
本文证明Bennewitz猜想，并结合计算Titchmarsh-Weyl矩阵在极点邻域Laurent展开的新算法，研究若干难以分析的微分方程的HELP不等式存在性。
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…
On a class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl--Titchmarsh theory
本文研究对应于矩阵弦方程的正则系统，利用Volterra算子与平方积分的线性相似性建立一般型基本解表示，并通过GBDT版本的Darboux变换构造显式解，同时发展相应的Weyl--Titchmarsh理论。
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…
On realization of the original Weyl-Titchmarsh functions by Shrödinger L-systems
该研究探讨原始Weyl-Titchmarsh函数通过薛定谔L-系统的实现问题，构造了对应不同边值问题的L-系统，推导了非负算子下的附加性质，给出了唯一性准则及主算子共享条件，并附示例说明。
The original Weyl-Titchmarsh functions and sectorial Shrödinger L-systems
本文研究非负极小对称薛定谔算子下原始Weyl-Titchmarsh函数生成的L-系统实现，推导其阻抗函数属于Stieltjes扇形类的充要条件，给出L-系统几何结构及算子扇形角异同的参数化描述，并附示例验证。
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 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.
The L-system representation and c-entropy
本文针对亏格指数为(1,1)的对称算子及其自伴扩张，构造唯一的L-系统使其阻抗映射与Weyl-Titchmarsh函数及其变换重合，并计算了所得系统的c-熵与主算子耗散系数。
Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices
本文用离散自伴Dirac系统替代Szegő递推与矩阵正交多项式来研究分块Toeplitz矩阵，证明Christoffel–Darboux公式的类似物并导出再生核渐近关系，进而刻画有理Weyl–Titchmarsh函数情形下出现的分块对角加分块半可分离Toeplitz矩阵。
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…
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…
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 matrix potentials on each edge. As inverse data we use the Weyl-Titchmarsh matrix function or the dynamic response operator.
We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.
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.
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 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.
Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions
本文研究广义规范系统中Titchmarsh-Weyl函数的高能渐进行为，引入矩阵形式的振幅函数新恒等式，并分析相应结构算子的性质。
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.
Boundary value problems for adjoint pairs of operators
该研究将拟边界三元组及其Weyl函数概念推广至算子伴随对框架，借助抽象Titchmarsh-Weyl $M$-函数得到边值问题唯一可解的充分条件并给出解的Krein型预解表示，理论可应用于一大类椭圆微分算子。
We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed G-manifold M, where G is a compact, connected Lie group acting effectively and isometrically on M. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it with the reduction of the corresponding Hamiltonian flow, proving that the equivariant spectral counting function satisfies Weyl's law, together with an estimate for the remainder.
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…
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.
Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law
本研究针对图上二阶Sturm-Liouville型微分算子的自伴延拓问题，建立了自伴性与哈密顿向量场非完整轨迹的关联，定义了顶点处的Kirchhoff条件以保证自伴延拓，并推导了谱渐近的Weyl定律。
We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and $2\times 2$ matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on $(a,\infty)$ with a potential strongly singular at the end point $a$.…
Dissipative eigenvalue problems for a singular quantum Dirac system
本文在特定Hilbert空间中研究耗散奇异q-Dirac算子，通过构造自伴膨胀等方法得到其散射矩阵、特征函数，并证明相关向量系的完备性。
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 study an inverse spectral problem for arbitrary order ordinary differential equations on compact star-type graphs when differential equations have regular singularities at boundary vertices. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.
We provide an abstract framework for singular one-dimensional Schroedinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove a new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Liebermann type result for these operators. Our approach is based on the singular Weyl-Titchmarsh theory which is extended to cover the present situation.
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 origina…
Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on $T^\ast M \times G$ with singular critical sets that were examined previously in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on $M$. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in $L^2(M)$. In forthcoming papers, the improved rema…
Eigenfunctions expansion for discrete symplectic systems with general linear dependence on spectral parameter
本文将对谱参数具有一般线性依赖的有限离散区间上离散辛系统的特征函数展开作为经典展开定理的重要推广，并利用合适的谱函数显式推导了Weyl--Titchmarsh $M(λ)$-函数的积分表示，同时探讨了半直线情形的推广。
Relations between half- and full-lattice CMV operators with scalar- and matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients results in a perturbation of twice the minimal rank. The explicit form for the minimal rank perturbation and the resulting two half-lattice CMV matrices are obtained. In addition, formulas relating the…
We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.
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