Borel summation of the semi-classical expansion of the partition function associated to the Schr\"odinger equation

TL;DR

This paper demonstrates that the partition function for a perturbed semi-classical harmonic oscillator can be reconstructed exactly from its divergent semi-classical series using Borel summation, bridging formal expansions and actual functions.

Contribution

It establishes the Borel summability of the semi-classical expansion of the partition function for perturbed harmonic oscillators, providing a rigorous link between formal series and the exact partition function.

Findings

Partition function is Borel summable.

Semi-classical expansion converges to the actual partition function.

Provides a rigorous mathematical foundation for semi-classical series.

Abstract

We prove that the partition function associated to a perturbation of the semi-classical harmonic oscillator is the Borel sum of its semi-classical expansion.