# Heat kernel for higher-order differential operators and generalized exponential functions

## TL;DR

This paper investigates the heat kernel for higher-derivative and nonlocal operators, revealing that it is expressed via generalized exponential functions related to Fox-Wright and Fox H-functions, with distinct asymptotic and oscillatory behaviors.

## Contribution

It introduces the use of generalized exponential functions to represent heat kernels of higher-derivative operators, challenging traditional exponential ansatz and enabling new analysis methods.

## Key findings

- Heat kernel expressed by generalized exponential functions (GEF).
- GEF exhibits oscillatory behavior unlike conventional exponential decay.
- Asymptotic and semiclassical expansions differ for local and nonlocal operators.

## Abstract

We consider the heat kernel for higher-derivative and nonlocal operators in $d$-dimensional Euclidean space-time and its asymptotic behavior. As a building block for operators of such type, we consider the heat kernel of the minimal operator - generic power of the Laplacian - and show that it is given by the expression essentially different from the conventional exponential Wentzel-Kramers-Brillouin (WKB) ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright $\varPsi$-functions and Fox $H$-functions. The structure of its essential singularity in the proper time parameter is different from that of the usual exponential ansatz, which invalidated previous attempts to directly generalize the Schwinger-DeWitt heat kernel technique to higher-derivative operators. In particular, contrary to the conventional exponential decay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivative operators. We give several integral representations for the generalized exponential function, find its asymptotics and semiclassical expansion, which turns out to be essentially different for local operators and nonlocal operators of noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.