Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Example Of Accurate Microlocal Tunneling In One Dimension

Antide Duraffour, Nicolas Raymond

TL;DR

This work develops a microlocal framework for a one-dimensional pseudo-differential Schrödinger-type operator that exhibits a symmetric double-well structure. By reducing the problem to an effective one-dimensional operator via WKB constructions and microlocalization, the authors derive sharp asymptotics for the first two eigenvalues: the gap satisfies $\lambda_2(\mathscr{L}_h) - \lambda_1(\mathscr{L}_h) \sim h(\lambda_2(\mathscr{M}_\hbar) - \lambda_1(\mathscr{M}_\hbar))$, with an exponentially small interaction term controlled by a phase $S$. The analysis combines the Kuranishi trick, Agmon-type weighted estimates, and stationary-phase arguments to produce optimal quasimodes, precise tunneling formulas, and rigorous exponential decay of eigenfunctions. These results extend the tunneling paradigm from differential to pseudo-differential operators and lay groundwork for future purely magnetic tunneling analyses in higher dimensions. The techniques offer a robust template for spectral analysis of PDOs with symmetric double-well symbols in the semiclassical limit.

Abstract

We investigate the spectral analysis of a class of pseudo-differential operators in one dimension. Under symmetry assumptions, we prove an asymptotic formula for the splitting of the first two eigenvalues. This article is a first example of extension to pseudo-differential operators of the tunneling effect formulas known for the symmetric electric Schr{ö}dinger operator.

An Example Of Accurate Microlocal Tunneling In One Dimension

TL;DR

This work develops a microlocal framework for a one-dimensional pseudo-differential Schrödinger-type operator that exhibits a symmetric double-well structure. By reducing the problem to an effective one-dimensional operator via WKB constructions and microlocalization, the authors derive sharp asymptotics for the first two eigenvalues: the gap satisfies , with an exponentially small interaction term controlled by a phase . The analysis combines the Kuranishi trick, Agmon-type weighted estimates, and stationary-phase arguments to produce optimal quasimodes, precise tunneling formulas, and rigorous exponential decay of eigenfunctions. These results extend the tunneling paradigm from differential to pseudo-differential operators and lay groundwork for future purely magnetic tunneling analyses in higher dimensions. The techniques offer a robust template for spectral analysis of PDOs with symmetric double-well symbols in the semiclassical limit.

Abstract

We investigate the spectral analysis of a class of pseudo-differential operators in one dimension. Under symmetry assumptions, we prove an asymptotic formula for the splitting of the first two eigenvalues. This article is a first example of extension to pseudo-differential operators of the tunneling effect formulas known for the symmetric electric Schr{ö}dinger operator.
Paper Structure (30 sections, 26 theorems, 187 equations, 2 figures)

This paper contains 30 sections, 26 theorems, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Framework, heuristics and main result
  4. Organization and strategy
  5. The one-well operator and WKB constructions
  6. Proof of Lemma \ref{['spdisc']}
  7. Proof of Proposition \ref{['prop.quasimodes']}
  8. WKB expansions
  9. Proof of Proposition \ref{['prop.quasimodes']}
  10. First tunneling estimate
  11. Microlocal harmonic approximation
  12. Microlocalization
  13. Proof of Proposition \ref{['prop.simplewell']}
  14. Proof of Proposition \ref{['prop.doublewell']}
  15. Localization
  16. ...and 15 more sections

Key Result

Theorem 1.4

Under Assumptions hyp:a, hyp:b and hyp:c, we have Moreover, for some $c>0$, $\lambda_3(\mathscr{L}_h)-\lambda_2(\mathscr{L}_h)\geqslant c h^{\frac{3}{2}}$.

Figures (2)

  • Figure 1: Sealing a well
  • Figure 2: The function $\chi_\ell$

Theorems & Definitions (44)

  • Theorem 1.4
  • Lemma 2.1
  • Proposition 2.2: WKB quasimodes
  • Remark 2.3
  • Proposition 2.4: WKB constructions
  • proof
  • Proposition 3.1: Microlocal harmonic approximation
  • Proposition 3.2: Rough tunneling estimate
  • Lemma 3.3: Localization of eigenfunctions
  • proof
  • ...and 34 more