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Existence of resonances for Schrodinger operators on hyperbolic space

David Borthwick, Yiran Wang

Abstract

We prove existence results and lower bounds for the resonances of Schrödinger operators associated to smooth, compactly support potentials on hyperbolic space. The results are derived from a combination of heat and wave trace expansions and asymptotics of the scattering phase.

Existence of resonances for Schrodinger operators on hyperbolic space

Abstract

We prove existence results and lower bounds for the resonances of Schrödinger operators associated to smooth, compactly support potentials on hyperbolic space. The results are derived from a combination of heat and wave trace expansions and asymptotics of the scattering phase.

Paper Structure

This paper contains 11 sections, 23 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. The resolvent
  3. Traces
  4. Birman-Krein formula
  5. Poisson formula
  6. Wave trace expansion
  7. Heat trace
  8. Scattering phase asymptotics
  9. Existence of resonances
  10. Even dimensions
  11. Odd dimensions

Key Result

Theorem 1.1

Let $\mathcal{R}_V$ denote the set of resonances of $\Delta + V$ for $V \in C^\infty_0(\mathbb{H}^{n+1},\mathbb{R})$, with $N_V(r)$ the corresponding counting function.

Theorems & Definitions (39)

  • Theorem 1.1
  • Proposition 3.1
  • Theorem 4.1: Birman-Krein formula
  • proof
  • Theorem 5.1: Poisson formula
  • Proposition 5.2
  • Proposition 5.3
  • proof : Proof of Theorem \ref{['poisson.thm']}
  • Theorem 6.1
  • Lemma 6.2
  • ...and 29 more