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Semiclassical study of shape resonances in the Stark effect

Kentaro Kameoka

Abstract

Semiclassical behavior of Stark resonances is studied. The complex distortion outside a cone is introduced to study resonances in any energy region for the Stark Hamiltonians with non-globally analytic potentials. The non-trapping resolvent estimate is proved by the escape function method. The Weyl law and the resonance expansion of the propagator are proved in the shape resonance model. To prove the resonance expansion theorem, the functional pseudodifferential calculus in the Stark effect is established, which is also useful in the study of the spectral shift function.

Semiclassical study of shape resonances in the Stark effect

Abstract

Semiclassical behavior of Stark resonances is studied. The complex distortion outside a cone is introduced to study resonances in any energy region for the Stark Hamiltonians with non-globally analytic potentials. The non-trapping resolvent estimate is proved by the escape function method. The Weyl law and the resonance expansion of the propagator are proved in the shape resonance model. To prove the resonance expansion theorem, the functional pseudodifferential calculus in the Stark effect is established, which is also useful in the study of the spectral shift function.

Paper Structure

This paper contains 14 sections, 30 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Definition of resonances
  3. Complex distortion
  4. Meromorphic continuations of matrix elements
  5. Non-trapping estimates
  6. Shape resonance model
  7. Resolvent estimate
  8. The Weyl law
  9. Resonance expansion
  10. Functional pseudodifferential calculus in the Stark effect
  11. Weighted resolvent estimates
  12. Weighted resolvents as $\Psi$DOs
  13. Proofs
  14. Commutator calculation

Key Result

Theorem 1

Suppose that Assumption 1 holds. Fix any $\hbar>0$. Then for any $\chi_1, \chi_2 \in L_{\mathrm{cone}}^{\infty}(\mathbb{R}^n)$ such that $\chi_j \not = 0$ on some open sets, the cutoff resolvent $\chi_1 R_{+}(z) \chi_2 \mspace{7mu} (\mathrm{Im}z>0)$ has a meromorphic continuation to $\mathrm{Im}z>-\ The set of resonances is independent of the choices of $\chi_1$ and $\chi_2$ including multipliciti

Theorems & Definitions (69)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1.1
  • Theorem 6
  • Corollary 1.1
  • Remark 1.2
  • Lemma 2.1
  • ...and 59 more