The high energy semiclassical asymptotics of loci of roots of fundamental solutions for polynomial potentials

TL;DR

This paper describes the semiclassical high-energy asymptotic distribution of roots of fundamental solutions to the 1-D Schrödinger equation with polynomial potentials, revealing their concentration along specific Stokes lines and island-like zero distributions.

Contribution

It provides a novel semiclassical analysis of root loci for fundamental solutions, detailing their distribution along exceptional Stokes lines in the high complex energy limit.

Findings

Roots of solutions concentrate on exceptional Stokes lines.

Loci form island-like structures escaping to infinity.

Finite roots connect pairs of turning points.

Abstract

In the case of polynomial potentials all solutions to 1-D Schroedinger equation are entire functions totally determined by loci of their roots and their behaviour at infinity. In this paper a description of the first of the two properties is given for fundamental solutions for the high complex energy limit when the energy is quantized or not. In particular due to the fact that the limit considered is semiclassical it is shown that loci of roots of fundamental solutions are collected of selected Stokes lines (called exceptional) specific for the solution considered and are distributed along these lines in a specific way. A stable asymptotic limit of loci of zeros of fundamental solutions on their exceptional Stokes lines has island forms and there are infintely many of such roots islands on exceptional Stokes lines escaping to infinity and a finite number of them on exceptional Stokes lines which connect pairs of turning points. The results obtained for asymptotic roots distributions of fundamental solutions in the semiclassical high (complex) energy limit are of a general nature for polynomial potentials.