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Magnetic Double-Wells: Lower Bounds on Tunneling

Charles L. Fefferman, Jacob Shapiro, Michael I. Weinstein

TL;DR

The paper proves that in magnetic double-well systems, generic couplings yield robust lower bounds on tunneling: the hopping coefficient $\\rho_0(\\lambda)$ and the ground-state splitting $\\Delta_0(\\lambda)$ are bounded below by $\\exp(-\\lambda^{1+\\varepsilon})$ outside a density-zero exceptional set. The authors achieve this by constructing an analytic continuation of the relevant quantities to a complex wedge, using a cut-off resolvent to circumvent non-analytic behavior of the Landau resolvent, and reducing the two-well problem to a $2\\times 2$ matrix problem whose eigenvalue splitting is controlled via a Blaschke-factor/Harmonic-majorant framework. A key technical contribution is a general complex-analytic lower-bound lemma that converts pointwise lower bounds together with global upper bounds into averaged lower bounds over real parameters. The results establish that vanishing tunneling is non-generic and quantify the extent to which generic couplings recover nonzero tunneling amplitudes, with implications for magnetic materials and mesoscopic systems. The work also lays out a detailed analytic apparatus—mesoscopic annuli, pseudodifferential calculus, and Landau-resolvent analysis—that could be adapted to higher Landau levels and more general magnetic settings.

Abstract

We present lower bounds on tunneling rates in magnetic double well systems for generic values of the coupling constant. This result was recently announced in \cite{FSW24} and complements our recent counter-example construction which exhibits vanishing tunneling for specially-constructed double-well potentials.

Magnetic Double-Wells: Lower Bounds on Tunneling

TL;DR

The paper proves that in magnetic double-well systems, generic couplings yield robust lower bounds on tunneling: the hopping coefficient and the ground-state splitting are bounded below by outside a density-zero exceptional set. The authors achieve this by constructing an analytic continuation of the relevant quantities to a complex wedge, using a cut-off resolvent to circumvent non-analytic behavior of the Landau resolvent, and reducing the two-well problem to a matrix problem whose eigenvalue splitting is controlled via a Blaschke-factor/Harmonic-majorant framework. A key technical contribution is a general complex-analytic lower-bound lemma that converts pointwise lower bounds together with global upper bounds into averaged lower bounds over real parameters. The results establish that vanishing tunneling is non-generic and quantify the extent to which generic couplings recover nonzero tunneling amplitudes, with implications for magnetic materials and mesoscopic systems. The work also lays out a detailed analytic apparatus—mesoscopic annuli, pseudodifferential calculus, and Landau-resolvent analysis—that could be adapted to higher Landau levels and more general magnetic settings.

Abstract

We present lower bounds on tunneling rates in magnetic double well systems for generic values of the coupling constant. This result was recently announced in \cite{FSW24} and complements our recent counter-example construction which exhibits vanishing tunneling for specially-constructed double-well potentials.

Paper Structure

This paper contains 60 sections, 40 theorems, 483 equations.

Table of Contents

  1. Introduction
  2. Rough sketch of the proof
  3. Consequences and outlook.
  4. Organization of this paper.
  5. Notation and conventions.
  6. Proof of the main theorem, \ref{['thm:main theorem']}
  7. The analyticity of the hopping coefficient $\lambda\mapsto\rho_0(\lambda)$
  8. The choice of $\mathbb{S}^1\ni\xi \mapsto z_\lambda(\xi)\in\mathbb C$ and $\Phi_\lambda$.
  9. Rationale for extending the resolvent.
  10. The cut-off Hamiltonian, $h_\lambda^\theta$, and the analytic extension of its resolvent
  11. The $\nu=0$ term: the MHO.
  12. The term $\nu=N_\Lambda$: the free Laplacian.
  13. The terms $\nu=1,\cdots,N_\Lambda-1$.
  14. Removing the cut-off to get from $h_\lambda^\theta$ to $h_\lambda$
  15. Intermediate conclusion.
  16. ...and 45 more sections

Key Result

Theorem 1.1

Let $v:\mathbb{R}^2\to[-1,0]$ be of class $C^3(\mathbb R^2)$ and supported within $B_a(0)$, where $a>0$. Further assume that $v(0)=-1$ is a unique global minimum which is non-degenerate. Let $\varepsilon>0$. Assume that $2d_1>Ca$ for some $C$ sufficiently large but independent of $\lambda$, where $d

Theorems & Definitions (74)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof : Proof of \ref{['lem:continuation of cut off one well resolvent']}
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof : Proof of \ref{['lem:continuation of one-well resolvent']}
  • Lemma 2.4
  • ...and 64 more