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Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

Brice Camus

TL;DR

The paper analyzes the semiclassical trace of Schrödinger operators at a critical energy where the potential has a totally degenerate maximum. It develops a local oscillatory integral framework around the equilibrium, deriving a complete asymptotic expansion for the fixed-point contribution, including potential logarithmic terms arising from an arithmetic condition on dimension and degeneracy. The leading term scales as h^{-n+n/2+n/(2k)} and involves invariant sphere-integrals of the leading homogeneous part V_{2k}, with logarithmic corrections determined by Mellin-residue analysis. This work extends semiclassical trace results to degenerate equilibria and provides a methodology for computing leading coefficients, with extensions to general h-admissible operators.

Abstract

We study the semi-classical trace formula at a critical energy level for a Schr\"odinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.

Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

TL;DR

The paper analyzes the semiclassical trace of Schrödinger operators at a critical energy where the potential has a totally degenerate maximum. It develops a local oscillatory integral framework around the equilibrium, deriving a complete asymptotic expansion for the fixed-point contribution, including potential logarithmic terms arising from an arithmetic condition on dimension and degeneracy. The leading term scales as h^{-n+n/2+n/(2k)} and involves invariant sphere-integrals of the leading homogeneous part V_{2k}, with logarithmic corrections determined by Mellin-residue analysis. This work extends semiclassical trace results to degenerate equilibria and provides a methodology for computing leading coefficients, with extensions to general h-admissible operators.

Abstract

We study the semi-classical trace formula at a critical energy level for a Schr\"odinger operator on . We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of and includes the singularity in . For these new contributions the asymptotic expansion involves the logarithm of the parameter . Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.

Paper Structure

This paper contains 6 sections, 13 theorems, 145 equations.

Table of Contents

  1. Introduction.
  2. Hypotheses and main result.
  3. Oscillatory representation of the spectral functions.
  4. Classical dynamic near the equilibrium.
  5. Normal forms of the phase function.
  6. Proof of the main result.

Key Result

Theorem 3

Under (H_{1}), if x_0 is a local minimum of the potential , homogeneous as in (H_3), then for all \varphi with \hat{\varphi}\in C_0^{\infty}(\mathbb{R}) we have : where the \Lambda _{j,l} are some computable distributions. The leading coefficient is : where \mathrm{S}(\mathbb{S}^{n-1}) is the surface of the unit-sphere of \mathbb{R}^n.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Remark 7
  • Lemma 8
  • Remark 9
  • Lemma 10
  • ...and 11 more