Finite Gap Potentials and WKB Asymptotics¶for One-Dimensional Schrödinger Operators

Abstract:

Consider the Schrödinger operator H=−d ²/dx ²+V(x) with power-decaying potential V(x)=O(x ^−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi inversion problem on hyperelliptic Riemann surfaces.