Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states

TL;DR

This paper extends path integral methods to complex eigenvalue problems in quantum physics, establishing a link between instantons and resonant states, and providing a new way to analyze metastable systems and $ ext{PT}$-symmetric scenarios.

Contribution

It generalizes the path integral approach to complex boundary conditions, enabling spectral analysis of non-Hermitian quantum systems and clarifying the instanton-resonance connection.

Findings

Identified the complex contour formulation for nonstandard boundary problems.

Established the correspondence between decay rates from tunneling and instantons.

Applied the method to resonant ground-state energy analysis.

Abstract

Schr\"odinger-type eigenvalue problems are ubiquitous in theoretical physics, with quantum-mechanical applications typically confined to cases for which the eigenfunctions are required to be normalizable on the real axis. However, seeking the spectrum of resonant states for metastable potentials or comprehending $\mathcal{PT}$-symmetric scenarios requires the broader study of eigenvalue problems for which the boundary conditions are provided in specific angular sectors of the complex plane. We generalize the conventional path integral treatment to such nonstandard boundary value problems, allowing the extraction of spectral information using functional methods. We find that the arising functional integrals are naturally defined on a complexified integration contour, encapsulating the demanded sectorial boundary conditions of the associated eigenvalue problem. The attained results are applied to the analysis of resonant ground-state energies, through which we identify the previously elusive one-to-one correspondence between decay rates derived from real-time quantum tunneling dynamics and those obtained via the Euclidean instanton method.