Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$

Abstract

We study Schrödinger operators with a one-center point interaction, formally defined by \begin{align*} -Δ_α=-Δ+α\,δ_0(\cdot), \end{align*} for $\alpha \in \mathbb{R}$, and the associated heat equation \begin{align} \partial_t u=\tfrac{1}{2}Δ_α u,\quad u(0,x)=u_0(x)\in C_c^{\infty}(\mathbb{R}^3\setminus\{0\}).\label{eq:HEapp} \end{align} Here $\Delta$ denotes the Laplacian (self-adjoint on $L^2({\mathbb{R}}^3)$) and ${\delta }_x$ the Dirac measure at $x$. The operator $-{\Delta }_{\alpha }$ can be realized either as a self-adjoint extension of $-\Delta {|}_{C_0^{\infty }({\mathbb{R}}^3\setminus \{0\})}$ in $L^2({\mathbb{R}}^3)$, or as the norm-resolvent limit of $-\Delta +{\lambda }_{\epsilon }V(\cdot /\epsilon )$ for suitable ${\lambda }_{\epsilon }$ and $V:{\mathbb{R}}^3\to \mathbb{R}$. In this paper we construct, for each $t>0$ and $x\in {\mathbb{R}}^3\setminus \{0\}$, a probability law on path space and a normalizing function $G_t^{\alpha }(x)$ giving the following probabilistic representation of the solution to the associated equation: \begin{align*} u(t,x)=G_t^α(x)\,\mathbb{E}\bigl[u_0\bigl(W^{t,x}(t)\bigr)\bigr], \end{align*} where $\{W^{t,x}(s):0\le s\le t\}$ is a continuous process depending on $(t,x,\alpha )$. The result provides a Feynman--Kac type formula for the heat equation with a one-point interaction in three dimensions.