Finite heat kernel expansions on the real line

Let $\mathcal{L}=d^2/dx^2+u(x)$ be the one-dimensional Schrödinger operator and let $H(x,y,t)$ be the corresponding heat kernel. We prove that the $n$th Hadamard's coefficient $H_n(x,y)$ is equal to 0 if and only if there exists a differential operator $\mathcal{M}$ of order $2n-1$ such that $\mathcal{L}^{2n-1}=\mathcal{M}^2$. 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 by Adler and Moser (1978) and Airault, McKean and Moser (1977). Equivalently, one can characterize the corresponding operators $\mathcal{L}$ as the rank one bispectral family given by Duistermaat and Grünbaum (1986).