A non-overdetermined inverse problem of finding the potential from the spectral function
Abstract
Description
Let $D\subset \R^n$, $n\geq 3,$ be a bounded domain with a $C^{\infty}$ boundary $S$, $L=-\nabla^2+q(x)$ be a selfadjoint operator defined in $H=L^2(D)$ by the Neumann boundary condition, $θ(x,y,λ)$ be its spectral function, $θ(x,y,λ):=\ds\sum_{λ_j<λ} ϕ_j(x)ϕ$ where $Lϕ_j=λ_jϕ_j$, $ϕ_{j N}|_S=0,$ $\|ϕ_j\|_{L^2(D)}=1$, $j=1,2,...$. The potential $q(x)$ is a real-valued function, $q\in C^\infty(D)$. It is proved that $q(x)$ is uniquely determined by the data $θ(s,s,λ) \forall s\in S$, $\forall λ\in \R_+$ if all the eigenvalues of $L$ are simple.
14pp
14pp