Random Matrices and Quantum Hamilton-Jacobi Method

TL;DR

This paper links the quantum Hamilton-Jacobi method with random matrix theory, showing that the evolution of complex poles in the Schrödinger equation can be described by quantum action and wave functions via random matrix distributions.

Contribution

It introduces a novel connection between quantum Hamilton-Jacobi approach and random matrix theory, providing a new perspective on quantum pole dynamics.

Findings

Complex pole evolution described by random matrices

Wave function represented by random matrix probability distribution

Connection to Cole-Hopf Transformation established

Abstract

In this paper, we start with the quantum Hamilton-Jacobi approach and show that the underlying complex pole evolution of the Schr\"odinger equation is described by the quantum action in terms of a random matrix. The wave function is given by the random matrix probability distribution function. In literature this is known as the famous Cole-Hopf Transformation.