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Ruijsenaars spectral transform

N. Belousov, S. Khoroshkin

Abstract

Spectral decomposition with respect to the wave functions of Ruijsenaars hyperbolic system defines an integral transform, which generalizes classical Fourier integral. For a certain class of analytical symmetric functions we prove inversion formula and orthogonality relations, valid for complex valued parameters of the system. Besides, we study four regimes of unitarity, when this transform defines isomorphisms of the corresponding $L_2$ spaces.

Ruijsenaars spectral transform

Abstract

Spectral decomposition with respect to the wave functions of Ruijsenaars hyperbolic system defines an integral transform, which generalizes classical Fourier integral. For a certain class of analytical symmetric functions we prove inversion formula and orthogonality relations, valid for complex valued parameters of the system. Besides, we study four regimes of unitarity, when this transform defines isomorphisms of the corresponding spaces.

Paper Structure

This paper contains 19 sections, 11 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Hamiltonians and wave functions
  4. Symmetries
  5. Bounds
  6. Spectral transform
  7. Inversion formula and orthogonality
  8. Statements
  9. Delta sequence
  10. Proof of Theorem \ref{['theoremf1']}
  11. Proof of Theorem \ref{['theoremf2']}
  12. Unitarity regimes
  13. Restrictions on parameters and scalar products
  14. Regimes I, II
  15. Regimes III, IV
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\bm{\lambda} \in \mathbb{R}^n$ and $\bm{x} \in \mathbb{C}^n$ such that Then for arbitrary $\delta > 0$

Theorems & Definitions (23)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3
  • Corollary 2
  • Lemma 1
  • proof
  • Proposition 2
  • ...and 13 more