Short-time asymptotics for the trace of one- and multi-dimensional Schrödinger semigroups
Author:
Vassilis G. Papanicolaou
Journal:
Proc. Amer. Math. Soc. 107 (1989), 927-935
MSC:
Primary 35P20; Secondary 35J10, 47D05, 47F05, 60J65
DOI:
https://doi.org/10.1090/S0002-9939-1989-0947315-1
MathSciNet review:
947315
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Abstract | References | Similar Articles | Additional Information
Abstract: Using Brownian motion we derive the leading asymptotic behaviour, as , of the (normalized) trace of
, where
is the operator
on
(with zero boundary condition at infinity),
is a hyperplane of
and
is the direct sum of
acting on
, with Dirichlet boundary condition on
(and 0 at infinity), and the same operator acting on
(
and
are the two half-spaces defined by
). The function
is taken bounded and continuous on
and, if
, we also assume that
is integrable on
(in fact we need a little less than that). We also show how to get higher order terms in our expansion, but in this case
is required to be smoother. In the one-dimensional case our result extends a result of Deift and Trubowitz (see the [D-T, Appendix]), since they proved a similar formula under the additional assumption that
as
.
The asympotic formula we give implies that can be recovered from certain spectral properties of
and
.
- [D-T] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979). MR 512420 (80e:34011)
- [P-W] M. Protter and H. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., 1967. MR 0219861 (36:2935)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1989-0947315-1
Keywords:
Trace formula,
Schrödinger semigroup,
(killed) Brownian motion
Article copyright:
© Copyright 1989
American Mathematical Society