On Spectral Properties of the Bloch--Torrey Operator in Two Dimensions

Abstract

We investigate a two-dimensional Schr\"odinger operator, $-h^2 \Delta +iV(x)$, with a purely complex potential $iV(x)$. A rigorous definition of this non-selfadjoint operator is provided for bounded and unbounded domains with common boundary conditions (Dirichlet, Neumann, Robin and transmission). We propose a general perturbative approach to construct its quasimodes in the semi-classical limit. An alternative WKB construction is also discussed. These approaches are local and thus valid for both bounded and unbounded domains, allowing one to compute the approximate eigenvalues to any order in the small $h$ limit. The general results are further illustrated on the particular case of the Bloch-Torrey operator, $-h^2\Delta + ix_1$, for which a four-term asymptotics is explicitly computed. Its high accuracy is confirmed by a numerical computation of the eigenvalues and eigenfunctions of this operator for a disk and circular annuli. The localization of eigenfunctions near the specific boundary points is revealed. Some applications in the field of diffusion nuclear magnetic resonance are discussed.