Heat kernel estimates for Schrödinger operators with supercritical killing potentials
Soobin Cho, Panki Kim, Renming Song
TL;DR
This work addresses sharp two-sided heat kernel and Green function estimates for the local Schrödinger operator $Δ-V$ with supercritical potentials near the origin, including inverse-power models $V(x)≈κ|x|^{-(2+2β)}$. It develops a probabilistic framework via the Feynman–Kac semigroup, introduces barrier/harmonic functions $h_{β,κ}$, $ ilde h_{β,κ}$, and $H_{d,β,κ}$, and exploits survival probabilities and exterior-domain heat kernel bounds to obtain comprehensive small-time and large-time bounds on $p(t,x,y)$ and $G(x,y)$. The paper proves the optimality of the local class ${\cal K}^{\rm loc}(β,κ)$ by showing failures of the estimates at the critical boundary, and provides explicit Green-function scaling that parallels the heat-kernel asymptotics. A key contribution is the explicit treatment of the purely local (non-fractional) case, contrasting with the fractional Schrödinger setting by revealing exponential near-origin decay and dimension-dependent large-time behavior. These results deepen understanding of spectral and probabilistic properties for Schrödinger operators with strong singular potentials and supply tools for applications in quantum mechanics and related fields.
Abstract
In this paper, we study the Schrödinger operator $Δ-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2β)}$, $b, β>0$. We obtain two-sided estimates on the heat kernel $p(t, x, y)$ of $Δ-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schrödinger operator $-(-Δ)^{α/2}-V$, $α\in (0, 2)$, with supercritical killing potential dealt with in [11], in the present case, the heat kernel $p(t, x, y)$ decays to 0 exponentially as $x$ or $y$ tends to the origin.