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

Abstract

We consider a non-self adjoint 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 $V\ge 0$, $V$ admitting a non-degenerate minimum at $x=0$ and ${\alpha }^{\prime }(0)=0$. We study its 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 $y$. In this paper, we show 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 $\{x=0\}$, and this scale is shown to be sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators.