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Sharp spectral gap for some degenerate Witten Laplacians

Loïs Delande

TL;DR

This work extends sharp Eyring–Kramers type descriptions from Morse to degenerate, non-M Morse Witten Laplacians in the semiclassical limit h→0. By developing sharp degenerate Gaussian quasimodes via an adapted WKB approach and constructing global geometric quasimode frameworks, the authors obtain precise low-lying eigenvalue asymptotics and explicit spectral gaps. A graded-matrix framework captures metastable transitions between wells, yielding EK-type formulas with explicit prefactors and exponents that depend on the local degeneracy data at saddle points. The results unify resolvent estimates, quasimode analysis, and a global interaction matrix, and are illustrated with admissible potentials and a triple-well example, highlighting the impact of degeneracy on metastable spectra. Overall, the paper broadens the scope of EK laws to degenerate potentials and provides tools applicable to degenerate Fokker–Planck operators and metastability analyses in higher dimensions.

Abstract

We consider Witten Laplacians associated to some non-Morse potentials. We prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the semiclassical regime and quantify the spectral gap separating these eigenvalues from the rest of the spectrum. The main ingredient is the construction of sharp degenerate Gaussian quasimodes through an adaptation of the WKB method.

Sharp spectral gap for some degenerate Witten Laplacians

TL;DR

This work extends sharp Eyring–Kramers type descriptions from Morse to degenerate, non-M Morse Witten Laplacians in the semiclassical limit h→0. By developing sharp degenerate Gaussian quasimodes via an adapted WKB approach and constructing global geometric quasimode frameworks, the authors obtain precise low-lying eigenvalue asymptotics and explicit spectral gaps. A graded-matrix framework captures metastable transitions between wells, yielding EK-type formulas with explicit prefactors and exponents that depend on the local degeneracy data at saddle points. The results unify resolvent estimates, quasimode analysis, and a global interaction matrix, and are illustrated with admissible potentials and a triple-well example, highlighting the impact of degeneracy on metastable spectra. Overall, the paper broadens the scope of EK laws to degenerate potentials and provides tools applicable to degenerate Fokker–Planck operators and metastability analyses in higher dimensions.

Abstract

We consider Witten Laplacians associated to some non-Morse potentials. We prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the semiclassical regime and quantify the spectral gap separating these eigenvalues from the rest of the spectrum. The main ingredient is the construction of sharp degenerate Gaussian quasimodes through an adaptation of the WKB method.

Paper Structure

This paper contains 15 sections, 16 theorems, 274 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Framework and results
  4. Resolvent and rough spectrum estimates
  5. Proof of Theorem \ref{['thm:1']}
  6. Sharp quasimodes
  7. Geometric constructions
  8. Proof of Theorem \ref{['thm:2']} and graded matrices
  9. Examples of admissible potentials
  10. Triple well, same height
  11. Some technical results
  12. Essential spectrum of $\Delta_V$
  13. Proof of Lemma \ref{['lem:soussolaffine']}
  14. Proof of Proposition \ref{['prop:saddle']}
  15. Degenerate Laplace method

Key Result

Lemma 1.1

Let $V$ satisfying Assumption ass.confin, then there exists $b\in\mathbb R$ such that with $C>0$ given by Assumption ass.confin.

Figures (2)

  • Figure 4.1: Representation of the potential $V$ near a point $\mathbf{s}\in\mathbf{j}(\mathbf{m})$. Here $H$ denotes the hypersurface defined by $\mathbf{s} + H = \{U^{-1}(x)_1 = \mathbf{s}_1\}$
  • Figure 6.1: Representation of a triple well function near its critical points.

Theorems & Definitions (31)

  • Lemma 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1
  • Proposition 1.5
  • Definition 1.6
  • Theorem 2
  • Remark 1.7
  • Proposition 2.1
  • ...and 21 more