On the heat trace of Schroedinger operators

Journal Article · Sun Dec 31 04:00:00 EST 1995 · Communications in Partial Differential Equations

DOI:https://doi.org/10.1080/03605309508821166· OSTI ID:482483

Banuelos, R; Sa Barreto, A ^[1]

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1. Purdue Univ., West Lafayette, IN (United States)

Trace formulae for heat kernels of Schroedinger operators have been widely studied in connection with spectral and scattering theory. They have been used to obtain information about a potential from its spectrum, or from its scattering data, and vice-versa. Using elementary Fourier transform methods we obtain a formula for the general coefficient in the asymptotic expansion of the trace of the heat kernel of the Schroedinger operator {minus}{Delta} + V, as t {down_arrow} 0, with V {element_of} S(R{sup n}), the class of functions with rapid decay at infinity. In dimension n = 1 a recurrent formula for the general coefficient in the expansion is obtained in [6]. However the KdV methods used there do not seem to generalize to higher dimension. Using the formula of [6] and the symmetry of some integrals, Y. Colin de Verdiere has computed the first four coefficients for potentials in three space dimensions. Also in [1] a different method is used to compute heat coefficients for differential operators on manifolds. 14 refs.

• Sponsoring Organization: USDOE
• OSTI ID: 482483
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 11-12 Vol. 20; ISSN CPDIDZ; ISSN 0360-5302
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ASYMPTOTIC SOLUTIONS
FOURIER TRANSFORMATION
KERNELS
QUANTUM OPERATORS
SCATTERING
SCHROEDINGER EQUATION