Analysis of the Fokker-Planck Equation in Schwarzschild Spacetime: A Supersymmetric Connection

TL;DR

This paper explores the connection between the Fokker-Planck equation in Schwarzschild spacetime and supersymmetric quantum mechanics, revealing a family of isospectral potentials with identical spectra but different wavefunctions.

Contribution

It establishes a novel link between Fokker-Planck dynamics in curved spacetime and SUSY QM, introducing isospectral deformations of the harmonic oscillator potential.

Findings

Fokker-Planck equation reduces to a harmonic oscillator form in Schwarzschild spacetime.

A family of isospectral potentials with the same energy spectrum but different wavefunctions is derived.

The supersymmetric framework provides new insights into the quantum dynamics in curved spacetime.

Abstract

We have re-analyzed the dynamics of the thermal potential within Schwarzschild spacetime by employing the Fokker-Planck equation. We demonstrate that the Fokker-Planck equation reduces to a simplified form equivalent to a scaled quantum mechanical problem with a harmonic oscillator potential. In this framework, we highlight an interesting correspondence between supersymmetric quantum mechanics (SUSY QM) and the Fokker-Planck dynamics associated with the Schwarzschild metric. Utilizing the isospectral deformation, an intrinsic feature of SUSY QM, we derive a family of one-parameter isospectral potentials. Notably, this new class of potentials exhibits the same energy spectrum as the original harmonic oscillator potential, but with distinct wavefunctions.