The semiclassical small-$\hbar$ limit of loci of roots of fundamental solutions for polynomial potentials

TL;DR

This paper analyzes the small-5 limit of zeros of fundamental solutions for polynomial potentials, showing roots cluster along Stokes lines in semiclassical regimes with various Stokes graph configurations.

Contribution

It provides a detailed description of the root loci behavior in the small-5 limit for different classes of polynomial potentials and Stokes graph structures.

Findings

Roots of fundamental solutions cluster along Stokes lines in the small-5 limit.

Infinitely many roots escape to infinity on Stokes lines, while finitely many remain on internal lines.

The analysis covers both quantized and non-quantized 5 regimes.

Abstract

In this paper a description of the small-$\hbar$ limit of loci of zeros of fundamental solutions for polynomial potentials is given. The considered cases of the potentials are bounded to the ones which provided us with simple turning points only. Among the latter potentials still several cases of Stokes graphs the potentials provide us with are distinguished, i.e. the general non-critical Stokes graphs, the general critical ones but with only single internal Stokes line and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized $\hbar$. In particular due to the fact that the small-$\hbar$ limit is semiclassical it is shown that loci of roots of fundamental solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of fundamental solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines.