On the spectral properties of the Bloch-Torrey equation in infinite periodically perforated domains

TL;DR

This paper studies the spectral and asymptotic behavior of the Bloch-Torrey operator in infinite periodically perforated domains, focusing on the spectrum's existence and its limits as the parameter g approaches infinity.

Contribution

It formalizes a numerical approach for analyzing the spectral properties of the Bloch-Torrey operator in complex domains and investigates its spectrum and asymptotics.

Findings

Existence of the spectrum established.

Asymptotic behavior characterized as g→∞.

Numerical methods validated for spectral analysis.

Abstract

We investigate spectral and asymptotic properties of the particular Schr\"odinger operator (also known as the Bloch-Torrey operator), $-\Delta + i g x$, in infinite periodically perforated domains of $\mathbb R^d$. We consider Dirichlet realizations of this operator and formalize a numerical approach proposed in [J. Phys. A: Math. Theor. 53, 325201 (2020)] for studying such operators. In particular, we discuss the existence of the spectrum of this operator and its asymptotic behavior as $g\to \infty$.