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On the spectral theory of sign-changing Laplace operators

Yves Colin de Verdière

TL;DR

This work develops a geometric, microlocal framework for the spectral theory of sign-changing Laplace operators across an interface $Z$ on a compact manifold $X$, leveraging semi-classical Dirichlet-to-Neumann maps to uncover interface-localized modes and resonances. By establishing ellipticity and self-adjointness for the transmission problem, deriving Weyl laws for both the positive and negative spectrum, and constructing semi-classical quasi-modes, the authors reveal how interface modes concentrate on $Z$ and provide a precise link between interface dynamics and the spectrum via a reduced operator $Q_h$. A central tool is the semi-classical DtN calculus, which yields a principal-symbol description and a mechanism to generate quasi-modes localized on the interface through a pseudo-differential operator $Q_h$ with symbol $q=\tfrac12 (k_+^* - k_-^*)-1$. In dimension two, the analysis recovers and extends known results (e.g., C-M) by giving explicit Weyl-type asymptotics for the interface spectrum in terms of the effective length of $Z$, thus connecting geometric interface data to spectral behavior with potential implications for metamaterials and wave localization.

Abstract

We study spectral theory of sign-changing Laplace operators using semi-classical Dirichlet-to-Neumann maps. We prove the existence of modesconcentrated on the interface and describe an effective semi-classical equation for them.

On the spectral theory of sign-changing Laplace operators

TL;DR

This work develops a geometric, microlocal framework for the spectral theory of sign-changing Laplace operators across an interface on a compact manifold , leveraging semi-classical Dirichlet-to-Neumann maps to uncover interface-localized modes and resonances. By establishing ellipticity and self-adjointness for the transmission problem, deriving Weyl laws for both the positive and negative spectrum, and constructing semi-classical quasi-modes, the authors reveal how interface modes concentrate on and provide a precise link between interface dynamics and the spectrum via a reduced operator . A central tool is the semi-classical DtN calculus, which yields a principal-symbol description and a mechanism to generate quasi-modes localized on the interface through a pseudo-differential operator with symbol . In dimension two, the analysis recovers and extends known results (e.g., C-M) by giving explicit Weyl-type asymptotics for the interface spectrum in terms of the effective length of , thus connecting geometric interface data to spectral behavior with potential implications for metamaterials and wave localization.

Abstract

We study spectral theory of sign-changing Laplace operators using semi-classical Dirichlet-to-Neumann maps. We prove the existence of modesconcentrated on the interface and describe an effective semi-classical equation for them.

Paper Structure

This paper contains 12 sections, 7 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Settings and summary of results
  3. Self-adjointness and ellipticity
  4. Weyl formulae
  5. Weyl for $M^2$
  6. Quasi-modes on $X_\varepsilon$
  7. Using minimax
  8. The semi-classical Dirichlet-to-Neumann maps
  9. Eigenfunctions localized on $Z$
  10. Existence of solutions of the equation $Q_hu_h=O(h^\infty )$
  11. Recovering the result of C-M for $d=2$
  12. Questions

Key Result

Theorem 3.1

If $({\bf Ell})$ is satisfied, the closure $\bar{M}$ of $M$ has domain $D\subset H^2(X_+)\oplus H^2(X_- )$ defined by the transmission condition $({\bf T })$ and $\bar{M}$ is self-adjoint. The spectrum of $M$ is discrete with eigenfunctions whose restrictions to $X_+$ and to $X_-$ are smooth.

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 7.1
  • Corollary 7.1