Further Theorems Inspired by Ambartsumian

We continue to derive results in the spirit of classical Ambartsumian Theorem 14.1 . In the first part we use heat kernel technique to show that a Schrödinger operator is isospectral to a Laplacian only if the potential is zero. This part is rather technical but does not require any a priori knowledge of heat kernel semigroups. In the second part the theory of almost periodic functions is used to obtain uniqueness results for Laplace and Schrödinger operators.