QLunch: Nicolas Raymond

Speaker: Nicolas Raymond, University of Angers

Title: Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane

Abstract: This talk is devoted to a non-selfadjoint operator of the form $-h^2 \Delta + i(V(x) + \alpha(x)y)$ on the upper half plane $y > 0$ with Dirichlet boundary conditions on $\{y = 0\}$ with $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$ and $\alpha'(0) = 0$.This operator appears when studying the so-called Bloch-Torrey equations, which are used to describe the magnetization diffusion of nuclei in a confined domain, and are the main model for the imaging technique known as diffusion MRI.

In this talk, we will focus on the eigenfunctions associated to the smallest eigenvalues in magnitude in the semiclassical limit $h \to 0$. Elementary variational estimates show that these eigenfunctions are localized near the point $(0,0)$ at the scales $O(h^{1/3})$ in $x$ and $O(h^{2/3})$ in $y$. We will see that the $O(h^{1/3})$ localization in $x$ is not optimal; more precisely, we establish that the eigenfunctions are concentrated in a neighborhood of size $O(h^{1/2})$ of the axis $\{x = 0\}$, which is sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators.

Joint work with M. Averseng, N. Frantz and F. Hérau.