Compactness of isospectral potentials

The Schrödinger operator , of a compact Riemannian manifold , has pure point spectrum. Suppose that is a smooth reference potential. Various criteria are given which guarantee the compactness of all satisfying . In particular, compactness is proved assuming an a priori bound on the norm of , where and . This improves earlier work of Brüning. An example involving singular potentials suggests that the condition is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions .