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

TL;DR

This paper investigates the localization properties of eigenfunctions of a non-self adjoint Bloch-Torrey operator on the half-plane, revealing sharper concentration scales near the minimum point than previously estimated.

Contribution

It demonstrates that eigenfunctions are localized in a neighborhood of size O(h^{1/2}) in x, improving upon the earlier O(h^{1/3}) estimate, and proves this scale is optimal.

Findings

Eigenfunctions concentrate near (0,0) at scale O(h^{1/2}) in x

The O(h^{1/3}) localization scale is not optimal and can be improved

The O(h^{1/2}) scale is shown to be sharp

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 $V \geq 0$, $V$ 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.