Cauchy Data for 1D singular Schr\"odinger operators

TL;DR

This paper develops methods to analyze semiclassical 1D Schrödinger operators with singular potentials, extending WKB solutions and deriving Bohr-Sommerfeld quantization rules with uniformity considerations.

Contribution

It introduces a novel approach to extend WKB expansions and compute Cauchy data for singular 1D Schrödinger operators, leading to new quantization rules.

Findings

Extended WKB expansions on [h^{1- ext{epsilon}},b]

Derived singular Bohr-Sommerfeld quantization rules

Achieved uniformity in WKB expansions with respect to W

Abstract

We study semiclassical 1-D Schr\"odinger operators of the form $Pu = -h^2 u'' \,+\,x^\gamma W(x) u$ on a finite interval $[0,b]$ for $0 < \gamma \in \mathbb{R} \setminus \mathbb{Q}$. We show that that the WKB expansions of solution can be extended on $[h^{1-\epsilon},b]$, for any $\epsilon>0$. Using a different approximation near $0$ and a matching procedure, we obtain the Cauchy Data at $0$ of such WKB solutions. This allows us to derive singular Bohr-Sommerfeld rules. We also pay special attention to uniformity in $W$ for our expansions.