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

• ### Denis S. Grebenkov

École Polytechnique, Palaiseau, France
• ### Bernard Helffer

Université de Nantes, France
• ### Nicolas Moutal

École Polytechnique, Palaiseau, France

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## Abstract

We investigate spectral and asymptotic properties of the particular Schrödinger 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 [17] for studying such operators. In particular, we discuss the existence of the spectrum of this operator and its asymptotic behavior as $g\to \infty$.