Perturbative expansions in quantum mechanics

TL;DR

This paper establishes an analytic deformation theorem for WKB expansions in quantum mechanics, demonstrating the analyticity of perturbation series for a harmonic oscillator after Borel transform, using local spectral definitions and Morse lemma.

Contribution

It introduces a D=1 analytic versal deformation theorem for WKB expansions and shows the analyticity of perturbation series via Borel transform techniques.

Findings

Proves an analytic deformation theorem for WKB expansions in one dimension.

Shows perturbation series become analytic after Borel transform.

Defines the spectrum of an operator in local analytic terms.

Abstract

We prove a D=1 analytic versal deformation theorem for WKB expansions. We define the spectrum of an operator in local analytic terms. We use the Morse lemma to show that the perturbation series arising in a perturbed harmonic oscillator become analytic after a formal Borel transform.