Finite heat kernel expansions on the real line

TL;DR

This paper characterizes when the heat kernel expansion for a 1D Schrödinger operator is finite, linking it to rational solutions of the KdV hierarchy and bispectral operators, revealing deep connections between heat kernel coefficients and integrable systems.

Contribution

It establishes a precise criterion for the finiteness of heat kernel expansions in terms of differential operators and rational KdV solutions, connecting spectral theory with integrable systems.

Findings

Heat kernel expansion is finite iff potential is a rational KdV solution.

Hadamard's coefficients vanish under a specific differential operator condition.

Finite expansion corresponds to rank one bispectral operators.

Abstract

Let L=d^2/dx^2+u(x) be the one-dimensional Schrodinger operator and H(x,y,t) be the corresponding heat kernel. We prove that the nth Hadamard's coefficient H_n(x,y) is equal to 0 if and only if there exists a differential operator M of order 2n-1 such that L^{2n-1}=M^2. Thus, the heat expansion is finite if and only if the potential u(x) is a rational solution of the KdV hierarchy decaying at infinity studied in [1,2]. Equivalently, one can characterize the corresponding operators L as the rank one bispectral family in [8].