Heat Kernel Expansion for Operators of the Type of the Square Root of the Laplace Operator

TL;DR

This paper introduces a generalized method for calculating DeWitt-Seeley-Gilkey coefficients for operators involving roots of Laplacians, extending to rational powers and simplifying calculations for positive operators.

Contribution

It proposes a novel pseudodifferential operator technique for operators with fractional powers, including explicit formulas linking coefficients with and without roots.

Findings

Calculated lowest DWSG coefficients for square root operators.

Extended method to operators with arbitrary rational powers.

Derived explicit formulas for DWSG coefficients of root operators.

Abstract

A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator $\sqrt{-\nabla^2 + V(x)}$ basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for the operator $\sqrt{-\nabla^2} + V(x)$ are calculated by using the method proposed. It is shown that the method admits a generalization to the case of operators of the type $(-\nabla^2 + V(X))^{1/{\rm m}}$, where m is an arbitrary rational number. A more simple method is proposed for the calculation of the DWSG coefficients for the case of strictly positive operators under the sign of root. By using this method, it is shown that the problem of the calculation of the DWSG coefficients for such operators is exactly solvable. Namely, an explicit formula expressing the DWSG coefficients for operators with root through the DWSG coefficients for operators without root is deduced.