Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions

Frédéric Hérau, David Krejcirik, Nicolas Raymond

Abstract

The Bloch--Torrey operator $-h^2Δ+e^{iα}x_1$ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming $α\in\left[0,\frac{3π}{5}\right)$ and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit $h \to 0$ are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.

Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions

Abstract

The Bloch--Torrey operator on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.
Paper Structure (24 sections, 32 theorems, 209 equations, 2 figures)

This paper contains 24 sections, 32 theorems, 209 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Heuristics
  4. The main result
  5. Consequences and extensions
  6. Organization and strategy
  7. The analytic dilation
  8. The sesquilinear form
  9. An isospectral operator
  10. Complex deformation parameters
  11. Localization estimates
  12. Spectral analysis
  13. Rough localization of the spectrum
  14. Resolvent estimates
  15. Quasimodes and localization estimates
  16. ...and 9 more sections

Key Result

Theorem 1.1

Assume $\alpha = 0$ and Assumption geoassumption0. Then, for all $n\geqslant 1$, as $h \to 0$, where $z_1$ is the absolute value of the smallest zero of the Airy function $\mathsf{Ai}$.

Figures (2)

  • Figure 1: The spectrum in the disk of center $z_1e^{\frac{2i\alpha}{3}}h^{\frac{2}{3}}$ and radius $Rh$, when $\alpha \in \left[0, \frac{3\pi}{5}\right)$.
  • Figure 2: Case when $\alpha\in\left(\frac{\pi}{2},\frac{3\pi}{5}\right)$.

Theorems & Definitions (55)

  • Theorem 1.1: CKPRS22
  • Theorem 1.2: AH16
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 45 more