  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 $g\to \infty$.
