Instantons and transseries of the Mathieu potential deformed by a $\mathcal{PT}$-symmetry parameter

TL;DR

This paper explores how a $ ext{PT}$-symmetry deformation affects the non-perturbative instanton solutions and transseries of the Mathieu potential, revealing modifications in instanton behavior and potential barriers with implications for quantum mechanics and Josephson junction models.

Contribution

It introduces a novel connection between non-Hermitian and Hermitian Mathieu systems via a complex reparameterization and analyzes the impact of $ ext{PT}$-symmetry deformation on instanton solutions and the potential landscape.

Findings

Deformation parameter smooths instanton passage between minima.

Potential barrier height decreases with increasing deformation.

Results extend to multi-instanton and bounce configurations.

Abstract

We investigate the non-perturbative effects of a deformation of the Mathieu differential equation consistent with $\mathcal{PT}$ symmetry. First, we develop a connection between the non-Hermitian and Hermitian scenarios by a reparameterization in the complex plane, followed by a restriction of the $\mathcal{PT}$ deformation parameter. The latter is responsible for preserving the information about $\mathcal{PT}$ symmetry when we choose to work in the Hermitian scenario. We note that this factor is present in all non-perturbative results and in the transseries representation of the deformed Mathieu partition function that we have obtained. In quantum mechanics, we found that the deformation parameter of $\mathcal{PT}$ symmetry has an effect on the real instanton solution for the deformed Mathieu potential in the Hermitian scenario. As its value increases, the non-Hermiticity factor makes it smoother for the instanton to pass from one minimum to another, that is, it modifies the instanton width. The explanation for this lies in the fact that the height of the potential barrier decreases as we increase the value of the deformation parameter. We present how this effect extends to the multi-instanton level and to the bounce limit of an instanton-anti-instanton pair. As an application of the obtained results, we show that the equation of motion under a tilted version of the potential in the Hermitian scenario compares to the resistively shunted junction (RSJ) model for the Josephson junction.