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Counter-examples to the fractal Weyl law for semiclassical resonances

Jean-Francois Bony, Setsuro Fujiie, Thierry Ramond, Maher Zerzeri

TL;DR

The paper shows that fractal Weyl upper bounds for semiclassical resonances are not generally sharp by constructing operators whose trapped sets have a prescribed dimension d in [1,n] but whose resonance count in a small energy window is uniformly bounded independently of h. The construction hinges on an escape function that isolates transport dynamics from resonance-generating trajectories, plus methods to remove specific heteroclinic connections (via absorbing potentials or higher-order/matrix-valued operators) without destroying the hyperbolic fixed points. It provides explicit realizations across several operator frameworks (Schrödinger with absorbing potential, fourth-order differential operators, and matrix-valued Schrödinger operators) and analyzes how the heteroclinic set can be tuned to produce trapped sets of arbitrary dimension. These counterexamples illuminate the nuanced role of trapped-set geometry in resonance distribution and caution against assuming fractal Weyl bounds are always sharp. The results leverage and extend existing semiclassical resonance techniques (escape functions, Sjöstrand–Zworski bounds, Helffer–Sjöstrand theory) to demonstrate the disconnect between fractal dimension and resonance counting in general settings.

Abstract

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.

Counter-examples to the fractal Weyl law for semiclassical resonances

TL;DR

The paper shows that fractal Weyl upper bounds for semiclassical resonances are not generally sharp by constructing operators whose trapped sets have a prescribed dimension d in [1,n] but whose resonance count in a small energy window is uniformly bounded independently of h. The construction hinges on an escape function that isolates transport dynamics from resonance-generating trajectories, plus methods to remove specific heteroclinic connections (via absorbing potentials or higher-order/matrix-valued operators) without destroying the hyperbolic fixed points. It provides explicit realizations across several operator frameworks (Schrödinger with absorbing potential, fourth-order differential operators, and matrix-valued Schrödinger operators) and analyzes how the heteroclinic set can be tuned to produce trapped sets of arbitrary dimension. These counterexamples illuminate the nuanced role of trapped-set geometry in resonance distribution and caution against assuming fractal Weyl bounds are always sharp. The results leverage and extend existing semiclassical resonance techniques (escape functions, Sjöstrand–Zworski bounds, Helffer–Sjöstrand theory) to demonstrate the disconnect between fractal dimension and resonance counting in general settings.

Abstract

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.

Paper Structure

This paper contains 14 sections, 5 theorems, 108 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Resonance upper bound
  3. The geometric setting
  4. Construction of an escape function
  5. End of the proof
  6. Examples
  7. The case of Schrödinger operators
  8. A Schrödinger operator with an absorbing potential
  9. The initial operator
  10. Removal of $\gamma_{2}$
  11. Heteroclinic set of dimension $2$
  12. Heteroclinic set of arbitrary dimension
  13. A differential operator of order $4$
  14. A matrix-valued Schrödinger operator

Key Result

Theorem 1.1

For any $d \in [1 , n]$, there exist operators $P = \mathop{\rm Op}\nolimits ( p )$ in settings generalizing a49--a15 and positive numbers $E_ {0} , \delta$ such that $i)$ the Hausdorff, Minkowski and packing dimensions of $K ( [ E_{0} - \delta , E_{0} + \delta ] )$ are precisely $d$. $ii)$ for any for $h$ small enough. $iii)$ there exists a smooth function $G ( x , \xi )$ on ${\mathbb R}^{2 n}$

Figures (6)

  • Figure 1: The trapped set $K ( E_{0} )$.
  • Figure 2: The potential of $Q$ and the set $K_{Q} ( E_{0} )$.
  • Figure 3: The cut-off functions $\chi_{\bullet}$ and the phase $\widetilde{\varphi}$.
  • Figure 4: The energy surface $q^{- 1} ( 0 ) \subset {\mathbb R}^{2}$.
  • Figure 5: The potentials $v_{1} , v_{2}$ on top and the energy surface $\det q = 0$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof