Short-time asymptotics for the trace of one- and multi-dimensional Schrödinger semigroups

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 ${L^H}$.