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Cardinal series representations for solutions of the Sturm–Liouville equation −y′′+q(x)y=ρ2y,x∈(0,L)$$ -{y}^{\prime \prime }+q(x)y={\rho}^2y,x\in \left(0,L\right) $$ with a complex‐valued potential q(x)$$ q(x) $$ are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to ρ$$ \rho $$ in any strip Imρ

In this paper, inverse problems are studied for Sturm–Liouville difference operators having $ \frac {p^{2}-1/4}{m^{2}} $ p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes. [ABSTRACT FROM AUTHOR]

In its original form the peak model for rank one supersingular perturbations of class ɦ--4 or higher of a nonnegative self-adjoint operator requires that the Gram matrix of the model should be diagonal. Here we remove the restriction on the Gram matrix. In particular we explain the origin of the Krein Q-function associated with the Gram matrix. [ABSTRACT FROM AUTHOR]

In this paper, we develop a systematic approach to treat Dirac operators A η , τ , λ with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths η , τ , λ ∈ ℝ , respectively, supported on points in ℝ , curves in ℝ 2 , and surfaces in ℝ 3 that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of A η , τ , λ . We make a substantial step towards more rough interaction supports Σ and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators A η , τ , λ are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of A η , τ , λ for arbitrary critical combinations of the interaction strengths under the condition that Σ is C ∞ -smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed. [ABSTRACT FROM AUTHOR]

We consider the indefinite Sturm–Liouville differential expression a (f) : = - 1 w 1 r f ′ ′ , where a is defined on a finite or infinite open interval I with 0 ∈ I and the coefficients r and w are locally summable and such that r(x) and (sgn x) w (x) are positive a.e. on I. With the differential expression a we associate a nonnegative self-adjoint operator A in the Krein space L w 2 (I) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space L | w | 2 (I) . These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces. [ABSTRACT FROM AUTHOR]

We provide the necessary and sufficient conditions for a generalized Nevanlinna function Q (Q ∊ Nκ (H)) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of Nκ(H), the functions that have a boundedly invertible derivative at infinity Q′ (∞):= limz→∞zQ(z). These functions are regular and have the operator representation Q(z)= ˜Γ+ (A-z)-1 ˜Γ, z ∊ ρ(A), where A is a bounded self-adjoint operator in a Pontryagin space K. We prove that every such strict function Q is a Weyl function associated with the symmetric operator S:= A|(I-P)K, where P is the orthogonal projection, P:= ˜Γ(˜Γ+ ˜Γ)-1 ˜Γ+. Additionally, we provide the relation matrices of the adjoint relation S+ of S, and of &#194, where &#194 is the representing relation of Q:= -Q-1. We illustrate our results through examples, wherein we begin with a given function Q ∊ Nκ (H) and proceed to determine the closed symmetric linear relation S and the boundary triple Π so that Q becomes the Weyl function associated with Π. [ABSTRACT FROM AUTHOR]

In this study, we consider the discontinuous Dirac equations system with eigenparameter dependent boundary and finite number of transmission conditions. First, the space that corresponds to problem is introduced, the norm on this space is defined and the operator model that corresponds to the given problem is constructed on this space. Then the integral equations and asymptotics of eigenfunctions of the problem are obtained. The characteristic function is defined and the asymptotic formula of the characteristic function is given by using obtained asymptotics of eigenfunctions. After the Weyl solution and the Weyl function of the problem are formed. Finally, some uniqueness theorems are proved by using Weyl function and some spectral data. [ABSTRACT FROM AUTHOR]

Abstract As early as 1910, Weyl gave a classification of the singular Sturm–Liouville equation, and divided it into the Limit Point Case and the Limit Circle Case at infinity. This led to the study of singular Sturm–Liouville spectrum theory. With the development of applications, the importance of singular Sturm–Liouville problems with a weighted function becomes more and more significant. This paper focuses on the study of singular Sturm–Liouville problems with a weighted function. Finally, an example of singular Sturm–Liouville problems with a weighted function is given.

Following the previous work, we shall study some inverse problems for the Dirac operator on an equilateral star graph. It is proven that the so-called Weyl function uniquely determines the potentials. Furthermore, we pay attention to the inverse problem of recovering the potentials from the spectral data, which consists of the eigenvalues and weight matrices, and present a constructive algorithm. The basic tool in this paper is the method of spectral mappings developed by Yurko. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse resonance problems for the discontinuous and non-selfadjoint Sturm-Liouville problem. We prove the uniqueness theorem and provide a reconstructive algorithm for the potential by using the Cauchy data and Weyl function. [ABSTRACT FROM AUTHOR]

We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter. Special cases of this problem include the recovery of the potential from the Weyl function, the inverse two-spectra Sturm–Liouville problem, as well as the recovery of the potential from the output boundary values of a plane wave that interacted with the potential. The method is based on the special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid, the problem is reduced to the classical inverse Sturm–Liouville problem of recovering the potential from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples. [ABSTRACT FROM AUTHOR]

In this work, we investigate the inverse problem for differential pencils with spectral boundary conditions having jump conditions on (0, 1). Taking the Weyl function technique, we prove a uniqueness theorem from the interior spectral data. [ABSTRACT FROM AUTHOR]

As early as 1910, Weyl gave a classification of the singular Sturm–Liouville equation, and divided it into the Limit Point Case and the Limit Circle Case at infinity. This led to the study of singular Sturm–Liouville spectrum theory. With the development of applications, the importance of singular Sturm–Liouville problems with a weighted function becomes more and more significant. This paper focuses on the study of singular Sturm–Liouville problems with a weighted function. Finally, an example of singular Sturm–Liouville problems with a weighted function is given. [ABSTRACT FROM AUTHOR]

The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$‐generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0. [ABSTRACT FROM AUTHOR]

The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.

We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter. Special cases of this problem include the recovery of the potential from the Weyl function, the inverse two-spectra Sturm–Liouville problem, as well as the recovery of the potential from the output boundary values of a plane wave that interacted with the potential. The method is based on the special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid, the problem is reduced to the classical inverse Sturm–Liouville problem of recovering the potential from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to the spectral data (the sequences of eigenvalues and normalized numbers) by using the Gelfand–Levitan–Marchenko method; the Weyl function. [ABSTRACT FROM AUTHOR]

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H . A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach. [ABSTRACT FROM AUTHOR]

Quantum system with spin–orbit interaction in three-dimensional space is considered. Point-like perturbation for Rashba Hamiltonian is constructed in the framework of the theory of self-adjoint extensions of symmetric operators. Boundary triplet approach is used. The Weyl function and the scattering matrix are constructed. [ABSTRACT FROM AUTHOR]

In this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, 0 < α ≤ 1 {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem. [ABSTRACT FROM AUTHOR]

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions. [ABSTRACT FROM AUTHOR]

The A-model for finite rank singular perturbations of class H-m-2∖H-m-1, m∈N, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces (Hn)n∈Z admit an orthogonal decomposition H-n⊕H+n, with the corresponding projections satisfying P±n+1⊆P±n, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces. [ABSTRACT FROM AUTHOR]

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter‐dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h1,h2, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H1,H2, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra. [ABSTRACT FROM AUTHOR]

With a closed symmetric operator A in a Hilbert space H a triple Π={H,Γ0,Γ1} of a Hilbert space H and two abstract trace operators Γ0 and Γ1 from A∗ to H is called a generalized boundary triple for A∗ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M(·) are investigated. The most important ones for applications are specific classes of boundary triples for which Green's second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on H, i.e. M(·)∈R(H), or at least they belong to the class R∼(H) of Nevanlinna families on H. The boundary condition Γ0f=0 determines a reference operator A0(=kerΓ0). The case where A0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1 admits a von Neumann type decomposition via A0 and the defect subspaces of A. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g. in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function M(·) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions M(·). Most involved ones concern operator functions M(·)∈R(H) for which τM(λ)(f,g)=(2iImλ)−1[(M(λ)f,g)−(f,M(λ)g)],f,g∈domM(λ),defines a closable nonnegative form on H. It turns out that closability of τM(λ)(f,g) does not depend on λ∈C± and, moreover, that the closure then is a form domain invariant holomorphic function on C± while τM(λ)(f,g) itself need not be domain invariant. In this study we also derive several additional new results, for instance, Kreĭn‐type resolvent formulas are extended to the most general setting of unitary and isometric boundary triples appearing in the present work. In part II of the present work all the main results are shown to have applications in the study of ordinary and partial differential operators. [ABSTRACT FROM AUTHOR]

In the present study, theorems related to the uniqueness of the solution of inverse problems for Dirac equations system are proved by applying spectral mapping method. With the help of this method, the inverse problem is reduced to the so-called main equation, which corresponds to the problem of existence and uniqueness of the solution of the system of linear equations in the Banach space. [ABSTRACT FROM AUTHOR]

We describe the Krein-von Neumann extension of minimal operator associated with the expression \(\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}\) on a finite interval \((a,b)\) in terms of boundary conditions. All non-negative extensions of the operator \(A\) as well as extensions with a finite number of negative squares are described.

In this work, we study boundary value problems for second-order differential equations with singularities of second kind inside an interval. First, we investigate properties of the spectrum, then prove a theorem on completeness of eigenfunctions, and study the inverse problem of reconstructing the differential equation from spectral characteristics. [ABSTRACT FROM AUTHOR]

We consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide. [ABSTRACT FROM AUTHOR]

We derive special representation for Weyl functions for finite and semi‐infinite Jacobi matrices with bounded entries based on a relationship between spectral problem for Jacobi matrices and initial‐boundary value problem for auxiliary dynamical systems with the discrete time. [ABSTRACT FROM AUTHOR]