Short-time asymptotics for the trace of one- and multi-dimensional Schrödinger semigroups
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- by Vassilis G. Papanicolaou PDF
- Proc. Amer. Math. Soc. 107 (1989), 927-935 Request permission
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 $|x| \to \infty$. The asympotic formula we give implies that $q$ can be recovered from certain spectral properties of $L$ and ${L^H}$.References
- P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
Additional Information
- © Copyright 1989 American Mathematical Society
- 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