The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (303 KB) | Metadata | Table of Contents | ZAA summary
Online access to the full text of Zeitschrift für Analysis und ihre Anwendungen is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
subscriptions@ems-ph.org
Volume 36, Issue 4, 2017, pp. 377–392
DOI: 10.4171/ZAA/1593

Published online: 2017-10-09

Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

Anna Canale[1], Abdelaziz Rhandi[2] and Cristian Tacelli[3]

(1) Università degli Studi di Salerno, Fisciano, Italy
(2) Università degli Studi di Salerno, Fisciano, Italy
(3) Università degli Studi di Salerno, Fisciano, Italy

We prove that the heat kernel associated to the Schrödinger type operator $A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}}$$ for $t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and $b=\frac{\beta-\alpha+2}{\beta+\alpha-2}$ provided that $N>2,\,\alpha\geq 2$ and $\beta>\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$.

Keywords: Schrödinger type operator, semigroup, heat kernel estimates

Canale Anna, Rhandi Abdelaziz, Tacelli Cristian: Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms. Z. Anal. Anwend. 36 (2017), 377-392. doi: 10.4171/ZAA/1593