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Finite heat kernel expansions on the real line

Plamen Iliev
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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].

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Plamen Iliev. 2005-04-14. Finite heat kernel expansions on the real line. https://doi.org/10.1090/s0002-9939-07-08677-7

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