Finite Heat Kernel Expansions on the Real Line

Let L = d/dx +u(x) be the one-dimensional Schrödinger operator and H(x, y, t) be the corresponding heat kernel. We prove that the nth Hadamard’s coefficient Hn(x, y) is equal to 0 if and only if there exists a differential operator M of order 2n− 1 such that L = M. 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].