Heat-kernel approach for scattering

TL;DR

This paper introduces a novel approach that leverages heat kernel and spectral methods from quantum field theory to solve scattering problems more effectively.

Contribution

It establishes a new method connecting heat kernel calculations with scattering problem solutions, using covariant perturbation theory.

Findings

Developed a heat-kernel based method for scattering problems.

Calculated off-diagonal heat-kernel expansion within covariant perturbation theory.

Provided an example linking heat kernels to scattering phase shifts.

Abstract

An approach for solving scattering problems, based on two quantum field theory methods, the heat kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of solving scattering problems. This allows us to establish a method of scattering problems from a method of heat kernels. As an application, we construct an approach for solving scattering problems based on the covariant perturbation theory of heat-kernel expansions. In order to apply the heat-kernel method to scattering problems, we first calculate the off-diagonal heat-kernel expansion in the frame of the covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partial-wave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.