Compactness of Isospectral Potentials
نویسنده
چکیده
The Schrödinger operator −∆+V , of a compact Riemannian manifold M , has pure point spectrum. Suppose that V0 is a smooth reference potential. Various criteria are given which guarantee the compactness of all V satisfying spec(−∆+V ) = spec(−∆+V0). In particular, compactness is proved assuming an a priori bound on the Ws,2(M) norm of V , where s > n/2 − 2 and n = dimM . 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 ≤ 9.
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