Inverse problem for singular diffusion operator

Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for…

Inverse spectral problems for Sturm-Liouville operators with singular potentials

The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space $W^{-1}_2(0,1)$. The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles

We develop the inverse scattering transform for the KdV equation with real singular initial data $q(x)$ of the form $q(x) = r'(x) + r(x)^2$, where $r\in L^2_{\textrm{loc}}$ and $r=0$ on $\mathbb R_+$. As a consequence we show that the solution $q(x,t)$ is a meromorphic function with no real poles for any $t>0$.

Inverse spectral problem for singular AKNS operator on [0,1]

We consider an inverse spectral problem for a class of singular AKNS operators $H\_a, a\in\N$ with an explicit singularity. We construct for each $a\in\N$, a standard map $λ^a\timesκ^a$ with spectral data $λ^a$ and some norming constant $κ^a$. For $a=0$, $λ^a\timesκ^a$ was known to be a local coordinate system on $\lr\times\lr$. Using adapted transformation operators, we extend this result to any non-negative integer $a$, give a description of isospectral sets and we obtain a Borg-Levinson type theorem.

Recovering Dirac systems with singularities in interior points

We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.

A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph

Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spectra. A uniqueness theorem is proved, and an algorithm for the constructive solution of the partial inverse problem is provided. The main ingredient of the proofs is the Riesz-basis property of specially constructed systems of functions.

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

In the paper, we study an inverse spectral problem for quadratic pencils of the Sturm--Liouville operators with singular coefficients and entire functions in the boundary conditions. We prove that a subspectrum is sufficient for recovering the pencil if this subspectrum generates a complete functional system. As well, we obtain an algorithm solving the inverse problem and alternative conditions on the subspectrum. Finally, these results are applied to studying a partial inverse problem.

An Inverse Problem for Sturm-Liouville Operators on the Half-line Having Bessel-type Singularity in an Interior Point

We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.

Half-inverse spectral problems for Sturm--Liouville operators with singular potentials

Half-inverse spectral problem for a Sturm--Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. We give the necessary and sufficient conditions for solvability of the half-inverse spectral problem for the class of Sturm--Liouville operators with singular potentials from the space $W^{-1}_2(0,1)$ and provide the reconstruction algorithm.

Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.

A half-inverse problem for the singular diffusion operator with jump conditions

In this paper, half inverse spectral problem for diffusion operator with jump conditions dependent on the spectral parameter and discontinuoty coeffcient is considered. The half inverse problems is studied of determining the coeffcient and two potential functions of the boundary value problem its spectrum by Hocstadt- Lieberman and Yang-Zettl methods. We show that two potential functions on the whole interval and the parameters in the boundary and jump conditions can be determined from spectrum.

Analyticity and uniform stability of the inverse singular Sturm--Liouville spectral problem

We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with real-valued distributional potentials in the Sobolev class W^{s-1}_2(0,1), s\in[0,1].

逆平方奇点与依赖于特征值参数的边界条件是同一事物的两个方面

Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin

本文证明逆平方奇点与含特征值参数的有理 Herglotz--Nevanlinna 函数边界条件等价，通过定义一维 Schrödinger 算子间的 Darboux 型变换，实现了两类边值问题谱结果的相互转换。

奇异径向AKNS算子的反谱分析

Inverse Spectral Analysis of Singular Radial AKNS Operators

本文研究了基于两种不同有效角动量参数κ值的奇异AKNS算子的反谱问题，重点分析零势附近的局部反问题，证明了部分参数对的局部唯一性。

Semiclassical inverse spectral theory for singularities of focus-focus type

We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.