Recovering Differential Operators with Nonlocal Boundary Conditions

Recovering Differential Operators on Spatial Networks

We give a short review of results on inverse spectral problems for ordinary differential operators on a spatial networks (geometrical graphs). We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential equations from spectral characteristics provided that the structure of the graph is known a priori. In the first half of the review we provide results related to inverse Sturm-Liouville problems on arbitrary compact graphs. Further, results on inverse problems for arbitrary order differential operators on compact graphs are presented. At…

Recovering Differential Operators with Nonseparated Boundary Conditions in the Central Symmetric Case

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

Recovering Higher Order Differential Systems with Regular Singularities on Star-type Graphs

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.

On recovering the Sturm--Liouville differential operators on time scales

We study Sturm--Liouville differential operators on the time scales consisting of a finite number of isolated points and segments. In a previous paper it was established that such operators are uniquely determined by their spectral characteristics. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm--Liouville boundary value problems with one common boundary condition alternate.

Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral matching condition. This operator is adjoint to the functional-differential operator with frozen argument at the central vertex of the graph. We study the inverse problem that consists in the recovery of the integral condition coefficients from the eigenvalues. We obtain the spectrum characterization, reconstruction algorithms, and prove the uniqueness of the i…

Inverse Nodal Problems for Dirac Type Integro Differential System with a Nonlocal Boundary Condition

In this work, the Dirac-type integro di§erential system with one classical boundary condition and another nonlocal integral boundary condition is considered. We obtain the asymptotic formulae for the solutions, eigenvalues and nodal points. We also investigate the inverse nodal problem and prove that given a dense subset of nodal points uniquely determines the boundary condition parameter and the potential function of the considered di§erential system. We also provide an e§ective procedure for solving the inverse nodal problem.

On recovering Sturm-Liouville operators with frozen argument from the spectrum

We consider second order linear differential operators possessing a term depending on the unknown function with a fixed argument and study the uniqueness of recovering the operators from the spectrum. We also obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability.

On the uniform stability of recovering sine-type functions with asymptotically separated zeros

We obtain a uniform stability of recovering entire functions of a special form from their zeros. To this form, one can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, and their non-local perturbations. We establish that the dependence of such functions on the sequences of their zeros possesses the Lipschitz property with respect to natural metrics on each ball of a finite radi…

Functional-differential operators on geometrical graphs with global delay and inverse spectral problems

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves {\it global} delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a {\it locally} nonlocal versio…

Recovering Variable Order Differential Operators with Regular Singularities on Graphs

We study inverse spectral problems for ordinary differential equations with regular singularities on compact star-type graphs when differential equations have different orders on diferent edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

关于恢复具有两个延迟的 Dirac 算子

On recovering Dirac operators with two delays

针对含两个常数延迟的 Dirac 型泛函微分算子，本文研究其逆谱问题，并给出由四个谱唯一恢复算子的延迟条件及不可能性边界。

非局部 $p$-拉普拉斯算子的变分特征值渐进行为研究

Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon

本文研究了具有可变视距的通用非局部 $p$-拉普拉斯算子的变分特征值渐进行为，利用 $\Gamma$-收敛论证，分析了视距变化极限情况下的解稳定性，恢复了局部 $p$-拉普拉斯和 $H^{s,p}$-拉普拉斯特征值问题的解。