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
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, https://doi.org/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
Keywords: Trace formula, Schrödinger semigroup, (killed) Brownian motion
Article copyright: © Copyright 1989 American Mathematical Society