Compactness of isospectral potentials

Source abstract

The Schrödinger operator − Δ + V -\Delta +V , of a compact Riemannian manifold M M , has pure point spectrum. Suppose that V 0 V_0 is a smooth reference potential. Various criteria are given which guarantee the compactness of all V V satisfying spec ⁡ ( − Δ + V ) = spec ⁡ ( − Δ + V 0 ) \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 ) W_{s,2}(M) norm of V V , where s > n / 2 − 2 s>n/2-2 and n = dim ⁡ M n=\dim M . This improves earlier work of Brüning. An example involving singular potentials suggests that the condition s > n / 2 − 2 s>n/2-2 is appropriate. Compactness is also proved for non–negative isospectral potentials in dimensions n ≤ 9 n\le 9 .