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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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 $ t \downarrow 0$, of the (normalized) trace of $ {e^{tL}} - {e^{t{L^H}}}$, where $ L$ is the operator $ \Delta /2 + q(x)$ on $ {{\mathbf{R}}^d}$ (with zero boundary condition at infinity), $ H$ is a hyperplane of $ {{\mathbf{R}}^d}$ and $ {L^H}$ is the direct sum of $ \Delta /2 + q(x)$ acting on $ {H^ + }$, with Dirichlet boundary condition on $ H$ (and 0 at infinity), and the same operator acting on $ {H^ - }$ ($ {H^ + }$ and $ {H^ - }$ are the two half-spaces defined by $ H$). The function $ q$ is taken bounded and continuous on $ {{\mathbf{R}}^d}$ and, if $ d \geq 2$, we also assume that $ q$ is integrable on $ {{\mathbf{R}}^d}$ (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 $ q$ 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 $ q(x) \to 0$ as $ \vert x\vert \to \infty $.

The asympotic formula we give implies that $ q$ can be recovered from certain spectral properties of $ L$ and $ {L^H}$.

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