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Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice

Jan Felipe van Diejen

Abstract

A quantum $n$-particle model consisting of an open $q$-difference Toda chain with two-sided boundary interactions is placed on a finite integer lattice. The spectrum and eigenbasis are computed by establishing the equivalence with a previously studied $q$-boson model from which the quantum integrability is inherited. Specifically, the $q$-boson-Toda correspondence in question yields Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall-Littlewood polynomials and provides the pertinent solutions of the Bethe Ansatz equations via the global minima of corresponding Yang-Yang type Morse functions.

Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice

Abstract

A quantum -particle model consisting of an open -difference Toda chain with two-sided boundary interactions is placed on a finite integer lattice. The spectrum and eigenbasis are computed by establishing the equivalence with a previously studied -boson model from which the quantum integrability is inherited. Specifically, the -boson-Toda correspondence in question yields Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall-Littlewood polynomials and provides the pertinent solutions of the Bethe Ansatz equations via the global minima of corresponding Yang-Yang type Morse functions.
Paper Structure (16 sections, 5 theorems, 64 equations)

This paper contains 16 sections, 5 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Open $q$-difference Toda chain with boundary interactions
  3. Quantum hamiltonian
  4. Proof of Proposition \ref{['sa:prp']}
  5. Eigenfunctions
  6. Bethe Ansatz
  7. Proof of Theorem \ref{['BA:thm']}
  8. Spectral analysis
  9. Solutions for the Bethe Ansatz equations
  10. Spectrum and eigenbasis
  11. The limit $q\to 1$
  12. Quantum hamiltonian
  13. Bethe Ansatz
  14. Solutions for the Bethe Ansatz equations
  15. Spectrum and eigenbasis for $q\to 1$
  16. ...and 1 more sections

Key Result

Proposition 2.1

For $\alpha_\pm\in (-1, 1)$, $\beta_\pm\in\mathbb{R}$, and $q\in (-1,1)\setminus \{ 0\}$, the $q$-difference Toda hamiltonian $H$Ht is self-adjoint in $\ell^2\bigl(\Lambda^{(n,m)},\Delta \bigr)$, i.e.

Theorems & Definitions (14)

  • Proposition 2.1: Self-adjointness
  • Remark 2.2
  • Theorem 3.1: Bethe Ansatz Wave Function
  • Remark 3.2
  • Proposition 4.1: Solutions for the Bethe Ansatz Equations
  • proof
  • Remark 4.2
  • Theorem 4.3: Spectrum and Eigenbasis
  • proof
  • Remark 4.4
  • ...and 4 more