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

TL;DR

The paper addresses the problem of localizing low-energy eigenfunctions of a non-selfadjoint Bloch-Torrey-type operator on the half-plane in the semiclassical limit. It develops a model reduction to a two-dimensional operator with a dominant transversal Airy component and uses operator-valued semiclassical pseudodifferential calculus to construct a parametrix around the first Airy mode, yielding an Agmon-type localization estimate. The main result shows that eigenfunctions concentrate near the axis $x=0$ at scale $O(h^{1/2})$ in the tangential direction, with the localization quantified by $oldsymbol{φ}_μ$ and proven to be sharp for $oldsymbol{μ}>0$. The paper also provides two crucial elliptic estimates on the projection onto the first Airy mode and its complement, alongside a detailed development of the operator-valued symbol calculus necessary for the 2D analysis. These results sharpen the understanding of localization phenomena for complex Bloch-Torrey operators and offer a rigorous framework applicable to diffusion MRI models and related physical problems.

Abstract

We consider a non-self adjoint operator of the form $-h^2 Δ+ i(V(x) + α(x)y)$ 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 $α'(0) = 0$. 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.

Figures (6)

• Figure 1: Plot of the magnitude of the $L^2$ normalized first eigenfunction of the Bloch-Torrey operator $-(2^{-n})^2\Delta + ix_1$ with Dirichlet boundary conditions on the two-dimensional domain $\Omega$ bounded by the curve parametrized by $r(\theta) = \cos(\theta)^3 + \sin(\theta)^3$ in polar coordinates, for $n = 7,8,9,10$ (from top-left to bottom-right). The red intensity is proportional to the magnitude of the plotted eigenfunction. Numerical computations were performed using the finite element method with a refined mesh near the point of the boundary where the eigenfunction is localized. The $L^2$ norm of the eigenfunction is indeed small outside of a box of size $O(h^{1/2})$ along the $y$-axis direction and $O(h^{2/3})$ along the $x$-axis, where $h = 2^{-n}$. The source code for these computations is available at code.
• Figure 2: Local parametrization of the boundary $\partial \Omega$ near the point of minimal abscissa.
• Figure 3: Plot of the magnitude of the $L^2$ normalized first eigenfunction of the operator $\mathscr{L}_h$ with $V = x^2$ and $\alpha = 1 - 0.1x^2$, truncated to a square domain $[-R,R] \times [0,2R]$ ($R = 4$) with Dirichlet boundary conditions on the boundary, and for $h = 2^{-n}$ with $n = 4,6,8,10$ (from top-left to bottom-right). The red intensity is proportional to the magnitude of the plotted eigenfunction. Numerical computations were performed using the finite element method with a refined mesh near near the axis $\{x = 0\}$. The $L^2$ norm of the eigenfunction is small outside a box of size $O(h^{1/2})$ along the $x$-direction and $O(1)$ along the $y$-direction. The source code for these computations is available at code.
• Figure 4: Graph of the magnitude of the eigenfunction computed numerically as in Figure \ref{['fig:numericsLh']} for $n = 10$ (left) and graph of the function $x \mapsto |\textup{Ai}(xe^{i \frac{\pi}{6}} + z_1)|$ (right).
• Figure 5: Poles of the resolvent $(\mathscr{A}_\alpha(x) - z)^{-1}$ (red), a region $\mathcal{P} \subset \mathbb{C}$ (blue hatched half plane) enclosing the set $U$ of the values taken by $h^{-2/3} \widetilde{\lambda}(x,\xi)$ for $(x,\xi) \in \mathbb{R}^2$ and $\lambda \in D(\lambda_{1,\alpha}(0)h^{2/3},Rh)$

Theorems & Definitions (71)

• Theorem 2.3
• Remark 2.4
• Remark 2.5: Removing the assumption on $\alpha$
• Definition 3.1: $\mu$-subsolution
• Proposition 3.2: Main elliptic estimate
• proof : Proof of Theorem \ref{['thm:main']} using Proposition \ref{['prop:main_ellip']}
• Proposition 3.3
• proof
• Definition 3.4: The operator $\Pi_{1,\alpha}$
• Lemma 3.5