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

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 $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$ and $\alpha'(0) = 0$. 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.