Multidimensional Borg-Levinson Theorem

Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues and the normal derivatives of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to $O(k^{-a})$ with $a>>1$, then we still have Hölder stability.

关于分数阶Borg-Levinson问题

On Fractional Borg-Levinson Problem

本文研究分数阶Borg-Levinson反谱问题，证明在容许类中势函数可由边界谱数据唯一确定，分数阶指数限于(1/2,1)，容许类仅依赖于区域和空间维度。

Stability for the multi-dimensional Borg--Levinson theorem of the biharmonic operator

In this paper, we prove a conditional Hölder stability estimate for the inverse spectral problem of the biharmonic operator. The proof employs the resolvent estimate and a Weyl-type law for the biharmonic operator which were obtained by the authors in \cite{LYZ}. This work extends nontrivially the result in \cite{stefanov} from the second order Schrödinger operator to the fourth order biharmonic operator.

A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data

We consider the multidimensional Borg-Levinson theorem of determining both the magnetic field $dA$ and the electric potential $V$, appearing in the Dirichlet realization of the magnetic Schrödinger operator $H=(-{\rm i}\nabla+A)^2+V$ on a bounded domain $Ω\subset\mathbb R^n$, $n\geq2$, from partial knowledge of the boundary spectral data of $H$. The full boundary spectral data are given by the set $\{(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ}):\ k\geq1\}$, where $\{ λ_k:\ k\in \mathbb N^* \}$ is the non-decreasing sequence of eigenvalues of $H$, $\{ ϕ_k:\ k\in \mathbb N^* \}$ an associated Hilberti…

Multidimensional Borg--Levinson theorems for unbounded potentials

We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-Δ+ q$, determine the potential $q$, when $q \in L^{n/2}(Ω,\mathbb{R})$ and $n \geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \in L^p(Ω,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$.

具有无界势的 Robin Laplacian 的多维 Borg-Levinson 唯一性与稳定性结果

Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential

针对三维及以上有界区域中带 Robin 边界条件的 Schrödinger 算子，本文利用特征值与完整或部分边界 Dirichlet 测量组成的谱数据，研究无界势反演的 Borg-Levinson 唯一性与稳定性。

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

We consider the multidimensional Borg-Levinson problem of determining a potential $q$, appearing in the Dirichlet realization of the Schrödinger operator $A_q=-Δ+q$ on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq2$, from the boundary spectral data of $A_q$ on an arbitrary portion of $\partialΩ$. More precisely, for $γ$ an open and non-empty subset of $\partialΩ$, we consider the boundary spectral data on $γ$ given by $\mathrm{BSD}(q,γ):=\{(λ_{k},{\partial_νϕ_{k}}_{|\overlineγ}):\ k \geq1\}$, where $\{ λ_k:\ k \geq1\}$ is the non-decreasing sequence of eigenvalues of $A_q$, $\{ ϕ_k:\ k \geq…

利用渐近边界谱数据确定双调和算子的一阶扰动

Determination of first order perturbation for bi-harmonic operator by asymptotic boundary spectral data

该研究针对有界光滑区域中的受扰双调和算子，证明可由 Dirichlet 特征值及特征函数 Neumann 迹的渐近边界谱数据稳定恢复其一阶与零阶系数。

Inverse problems for Schrödinger equations with unbounded potentials

We summarize in these notes the course given at the Summer School of AIP 2019 held in Grenoble from July 1st to July 5th. This course was mainly devoted to the determination of the unbounded potential in a Schrödinger equation from the associated Dirichlet-to-Neumann map (abbreviated to DN map in this text). We establish a stability inequality for potentials belonging to $L^n$, where $n\ge 3$ is the dimension of the space. Next, we prove a uniqueness result for potentials in $L^{n/2}$, $n\ge 3$, and apply this uniqueness result to demonstrate a Borg-Levinson type theorem. We use a classical ap…