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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finite heat kernel expansions on the real line


Author: Plamen Iliev
Journal: Proc. Amer. Math. Soc. 135 (2007), 1889-1894
MSC (2000): Primary 35Q53, 37K10, 35K05
Published electronically: January 8, 2007
MathSciNet review: 2286101
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Abstract: 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).


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Additional Information

Plamen Iliev
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160
Email: iliev@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08677-7
PII: S 0002-9939(07)08677-7
Received by editor(s): January 16, 2006
Received by editor(s) in revised form: February 23, 2006
Published electronically: January 8, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society