Poles of Integrale Tritronquee and Anharmonic Oscillators. A WKB Approach

TL;DR

This paper explores the relationship between poles of Painleve-I solutions and cubic anharmonic oscillators using a complex WKB method, providing insights into their quantization conditions.

Contribution

It introduces a WKB-based analysis of the quantization conditions linking Painleve-I poles and anharmonic oscillators, advancing the understanding of their mathematical connection.

Findings

Poles of Painleve-I correspond to specific cubic oscillators.

The complex WKB method effectively analyzes the quantization conditions.

Results clarify the structure of integrale tritronquee poles.

Abstract

Poles of solutions to the Painleve-I equation are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrale tritronquee are in 1-1 correspondence with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We analyze this pair of quantization conditions by means of a suitable version of the complex WKB method.