Eigenvalues and eigenfunctions of Schrodinger operators: Inverse spectral theory; and the zeros of eigenfunctions.

摘要

This dissertation contains two disjoint parts:;Part I. In the first part (which is from [H1]) we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrodinger operator. As an application, we prove similar inverse spectral results, obtained by Guillemin and Uribe in [GU], using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V( x).;Part II. In the second part (which is from [H2]) we study the semi-classical distribution of the complex zeros of the eigenfunctions of the 1D Schrodinger operators for the class of real polynomial potentials of even degree, with fixed energy level, E. We show that as hn → 0 the zeros tend to concentrate on the union of some level curves reals(S(zm, z)) = cm where S( zm, z) = zmz Vt-E dt is the complex action, and zm is a turning point. We also calculate these curves for some symmetric and non-symmetric one-well and double-well potentials.