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

MathSciNet review:
947315

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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), no. 2, 121–251. MR**512420**, 10.1002/cpa.3160320202**[P-W]**Murray H. Protter and Hans F. Weinberger,*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0947315-1

Keywords:
Trace formula,
Schrödinger semigroup,
(killed) Brownian motion

Article copyright:
© Copyright 1989
American Mathematical Society