Related papers: Survey of the inverse spectral problem
We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three…
We review basic ideas and basic examples of the theory of the inverse spectral problems.
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
In this paper we consider determining a minimal surface embedded in a Riemannian manifold $\Sigma\times \mathbb{R}$. We show that if $\Sigma$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated…
We consider the problem of reconstruction of the Cauchy data for the wave equation in $\mathbb{R}^3$ and $\mathbb{R}^2$ by the measurements of its solution on the boundary of the unit ball.
This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main…
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results…
Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with…
In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a…
The inverse problem of finding the refraction coefficient of a shallow ocean from the observation of the extra Cauchy data for the acoustic field at the surface of the ocean is studied. Uniqueness theorem is proved and a reconstruction…
Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results, due to the author are presented. The classical results are often presented in a new way. Several highlights…
In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave…
Different practical problems, espesially, problems of hydrodynamics, elasticity theory, geophysics and aerodynamics can be reduced to finding of an optimal shape. The investigation of these problems is based on the study of depending domain…
We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and…
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions…
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…
This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the…
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis…
We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
Interest in inverse dynamical, spectral and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum…