具有部分谱数据的磁薛定谔算子的多维Borg-Levinson定理
A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data
摘要：
我们认为多维Borg-Levinson定理确定磁场$dA$和电势$V$，出现在Dirichlet实现的磁薛定谔算子$H=(-{\rm i}\nabla+A)^2+V$上的有界域$Ω\subset\mathbb R^n$，$n\geq2$，从部分知识的边界谱数据的$H$。全边界光谱数据由集合$\{(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ}):\ k\geq1\}$给出，其中$\{ λ_k:\ k\in \mathbb N^* \}$是$H$的特征值的非递减序列，$\{ ϕ_k:\ k\in \mathbb N^* \}$是特征函数的相关希尔伯特基，$ν$是$\partialΩ$的单位向外法向量。我们证明了$(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ})$关于$k\geq1$的一些渐近知识唯一地确定了磁场$dA$和电势$V$。
英文摘要：
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 Hilbertian basis of eigenfunctions and $ν$ is the unit outward normal vector to $\partialΩ$. We prove that some asymptotic knowledge of $(λ_{k},{\partial_νϕ_{k}}_{|\partialΩ})$ with respect to $k\geq1$ determines uniquely the magnetic field $dA$ and the electric potential $V$.
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