  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].
