Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral estimates for degenerate critical levels

Brice Camus

TL;DR

This work derives semi-classical spectral asymptotics for $h$-pseudodifferential operators at a totally degenerate critical energy. Using a trace formula and a Fourier integral operator framework, the authors resolve the total degeneracy with geometrically meaningful normal forms obtained via a blow-up near the equilibrium, yielding a full asymptotic expansion for the localized trace $\gamma_{z_0}(E_c,\varphi,h)$. The leading term scales as $h^{\frac{2n}{k}-n}$ and is expressed invariantly through the Liouville measure on the critical set $C(\mathfrak p_k)$, with logarithmic corrections when $\frac{2n}{k}$ is an integer; an integrable singularity recovers a Weyl-type law. The results hold in any dimension and illuminate how non-integrable singularities amplify spectral contributions, linking spectral data to the geometry of singularities on the energy surface.

Abstract

We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. $h$ of the spectral distributions are bigger than in the regular setting.

Spectral estimates for degenerate critical levels

TL;DR

This work derives semi-classical spectral asymptotics for -pseudodifferential operators at a totally degenerate critical energy. Using a trace formula and a Fourier integral operator framework, the authors resolve the total degeneracy with geometrically meaningful normal forms obtained via a blow-up near the equilibrium, yielding a full asymptotic expansion for the localized trace . The leading term scales as and is expressed invariantly through the Liouville measure on the critical set , with logarithmic corrections when is an integer; an integrable singularity recovers a Weyl-type law. The results hold in any dimension and illuminate how non-integrable singularities amplify spectral contributions, linking spectral data to the geometry of singularities on the energy surface.

Abstract

We establish spectral estimates at a critical energy level for -pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. of the spectral distributions are bigger than in the regular setting.

Paper Structure

This paper contains 6 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction.
  2. Hypotheses and main result.
  3. Oscillatory representation.
  4. Classical dynamic near the equilibrium.
  5. Normal forms and oscillatory integrals.
  6. Proof of the main result.

Key Result

Theorem 4

Under hypotheses (H_{1}) to (H_{4}) we obtain the existence of a full asymptotic expansion : where the logarithms only occur when (2n+j)/k\in \mathbb{N}^* and \Lambda_{j,m}\in \mathcal{S}'(\mathbb{R}). As concerns the leading term we obtain : (1) If k>2n (non-integrable singularity on \Sigma_{E_c}) we have : where the distributional coefficient \Lambda _{0,0}(\varphi ) is given by : (2) If the

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Corollary 6
  • Corollary 7
  • Lemma 8
  • Theorem 9
  • Remark 10
  • ...and 16 more