We prove short time estimates for the heat kernel of Schrödinger operators with unbounded potential in $R^N$. More precisely, if we denote by $p(x; y; t)$ the heat kernel of the Schrödinger operator $H =-\Delta +V$, then we prove upper bounds like $p(x; y; t)\le c(t)\phi(x)\phi(y)$ for a large class of potentials tending to $+\infty$ as $|x| \to \infty$ , under the main assumption that $\omega =1/\phi$ satisfies $\omega(x)\to +\infty$ as $|x|\to \infty$ and $H\omega \ge g o \omega$ , where g is a convex function growing faster than linearly. The behaviour of c(t) near 0 is also shown to be precise. Similar bounds are also proved for the derivatives of p. Our analysis provides a family of such estimates e.g. for $V(x)=|x|^\alpha$ for every $\alpha >0$.

Kernel Estimates for Schroedinger Operators

RHANDI, Abdelaziz
2006-01-01

Abstract

We prove short time estimates for the heat kernel of Schrödinger operators with unbounded potential in $R^N$. More precisely, if we denote by $p(x; y; t)$ the heat kernel of the Schrödinger operator $H =-\Delta +V$, then we prove upper bounds like $p(x; y; t)\le c(t)\phi(x)\phi(y)$ for a large class of potentials tending to $+\infty$ as $|x| \to \infty$ , under the main assumption that $\omega =1/\phi$ satisfies $\omega(x)\to +\infty$ as $|x|\to \infty$ and $H\omega \ge g o \omega$ , where g is a convex function growing faster than linearly. The behaviour of c(t) near 0 is also shown to be precise. Similar bounds are also proved for the derivatives of p. Our analysis provides a family of such estimates e.g. for $V(x)=|x|^\alpha$ for every $\alpha >0$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/2295269
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