相关论文: Inverse spectral problems on a closed manifold

We consider the inverse problem of determining the metric-measure structure of collapsing manifolds from local measurements of spectral data. In the part I of the paper, we proved the uniqueness of the inverse problem and a continuity…

In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schr\"odinger operators, Laplacians on…

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show…

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $\Gamma_1$ and $\Gamma_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted…

We consider the inverse problem of the reconstruction of a Schr\"odinger operator on a unknown Riemannian manifold or a domain of Euclidean space. The data used is a part of the boundary $\Gamma$ and the eigenvalues corresponding to a set…

We consider a strongly damped wave equation on compact manifolds, both with and without boundaries, and formulate the corresponding inverse problems. For closed manifolds, we prove that the metric can be uniquely determined, up to an…

The classical Gel'fand's inverse problem asks whether a Riemannian manifold is uniquely determined by the knowledge of the heat kernel on any open subset of the manifold. We study this inverse problem in the non-smooth setting in the…

Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by…

We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension $n \ge 3$. We show that a continuous first order perturbation can be…

We consider a smooth Riemannian metric tensor $g$ on $\R^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - \Delta_g u = F$. Here, $F=F(t,x,\omega)$ is a random source that has white noise distribution…

We consider a compact Riemannian manifold $M$ (possibly with boundary) and $\Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $\Sigma$ for both the wave and heat…

We address the question of whether a Riemannian manifold-with-boundary (M,g) in dimension two is uniquely determined from knowledge of the distances between points on its boundary. An affirmative answer is called boundary rigidity for…

We show that on a simple Riemannian manifold, the electric potential and the solenoidal part of the magnetic potential appearing in the magnetic Schr\"odinger operator can be recovered H\"older stably from the boundary spectral data. This…

In this paper, we study the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data. Our contributions are twofold: first we introduce a Dirichlet-to-Neumann map for this operator and analyze an associated inverse problem;…

We formulate an inverse problem for an uncoupled space-time fractional Schr\"odinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the…

In this paper, we study the direct and inverse spectral problems for the Schrodinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic…

We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=B\cup G$, where $B$ is called the set of the…

We study the inverse spectral problem of jointly recovering a radially symmetric Riemannian metric and an additional coefficient from the Dirichlet spectrum of a perturbed Laplace-Beltrami operator on a bounded domain. Specifically, we…

The problem of obtaining the lower bounds on the restriction of Laplacian eigenfunctions to hypersurfaces inside a compact Riemannian manifold $(M,g)$ is challenging and has been attempted by many authors \cite{BR, GRS, Jun, ET}. This paper…

We study the inverse problem of unique recovery of a complex-valued scalar function $V:\mathcal M \times \mathbb C\to \mathbb C$, defined over a smooth compact Riemannian manifold $(\mathcal M,g)$ with smooth boundary, given the Dirichlet…