A non-overdetermined inverse problem of finding the potential from the spectral function

Abstract

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, $\theta(x,y,\lambda)$ be its spectral function, $\theta(x,y,\lambda):=\ds\sum_{\lambda_j<\lambda} \phi_j(x)\phi$ where $L\phi_j=\lambda_j\phi_j$, $\phi_{j N}|_S=0,$ $\|\phi_j\|_{L^2(D)}=1$, $j=1,2,...$. 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 $\theta(s,s,\lambda) \forall s\in S$, $\forall \lambda\in \R_+$ if all the eigenvalues of $L$ are simple.