Compactness of isospectral potentials

The Schrödinger operator $-\Delta+V$, of a compact Riemannian manifold $M$, has pure point spectrum. Suppose that $V_0$ is a smooth reference potential. Various criteria are given which guarantee the compactness of all $V$satisfying $\operatorname{spec}(-\Delta+V)=\operatorname{spec}(-\Delta+V_0)$. In particular, compactness is proved assuming an a priori bound on the $W_{s,2}(M)$ norm of $V$, where $s>n/2-2$ and $n=\dim M$. This improves earlier work of Brüning. An example involving singular potentials suggests that the condition $s>n/2-2$ is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions $n\le 9$.