Fundamental solutions of generalized non-local Schrodinger operators

Let $d \in {\lbrace 1, 2, 3, \dotsc \rbrace}$ and $s \in (0, 1)$ be such that $d \gt 2s$. We consider a generalized non-local Schrodinger operator of the form\[L=L_K + \nu \; \textrm{,}\]where $L_K$ is a non-local operator with kernel $K$ that includes the fractional Laplacian $(-\Delta)^s$ for $s \in (0, 1)$ as a special case. The potential $\nu$ is a doubling measure subjected to a certain constraint. We show that the fundamental solution of $L$ exists, is positive and possesses extra decaying properties.

Keywords

generalized fractional Schrodinger operator, fundamental solution, off-diagonal estimates

2010 Mathematics Subject Classification

35D30, 35J60, 45K05, 65M80

Full Text (PDF format)

Received 6 March 2021

Received revised 19 July 2021

Accepted 2 August 2021

Published 16 May 2022