Finite heat kernel expansions on the real line
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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).References
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Additional Information
- Plamen Iliev
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160
- MR Author ID: 629581
- Email: iliev@math.gatech.edu
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1889-1894
- MSC (2000): Primary 35Q53, 37K10, 35K05
- DOI: https://doi.org/10.1090/S0002-9939-07-08677-7
- MathSciNet review: 2286101