Quantum Mechanics with Random Imaginary Scalar Potential

TL;DR

This paper investigates the spectral properties of a non-Hermitian quantum Hamiltonian with a random imaginary potential, deriving analytical results for Green functions, eigenvalue distributions, and polymer chain statistics in higher dimensions.

Contribution

It introduces an effective field theory approach to analyze non-Hermitian quantum systems with randomness and connects quantum spectral properties to polymer statistical mechanics.

Findings

Derived an analytical expression for the averaged Green function.

Obtained the density distribution of complex eigenvalues.

Determined the polymer distribution function in high dimensions.

Abstract

We study spectral properties of a non-Hermitian Hamiltonian describing a quantum particle propagating in a random imaginary scalar potential. Cast in the form of an effective field theory, we obtain an analytical expression for the ensemble averaged one-particle Green function from which we obtain the density of complex eigenvalues. Based on the connection between non-Hermitian quantum mechanics and the statistical mechanics of polymer chains, we determine the distribution function of a self-interacting polymer in dimensions $d>4$.