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BC Toda chain I: reflection operator and eigenfunctions

N. Belousov, S. Derkachov, S. Khoroshkin

Abstract

We obtain Gauss-Givental integral representation for the eigenfunctions of quantum Toda chain with boundary interaction of BC type. For this we introduce reflection operator satisfying reflection equation with DST chain Lax matrices. Besides, we define Baxter operators for BC Toda chain, prove their commutativity with Hamiltonians and derive the corresponding Baxter equation.

BC Toda chain I: reflection operator and eigenfunctions

Abstract

We obtain Gauss-Givental integral representation for the eigenfunctions of quantum Toda chain with boundary interaction of BC type. For this we introduce reflection operator satisfying reflection equation with DST chain Lax matrices. Besides, we define Baxter operators for BC Toda chain, prove their commutativity with Hamiltonians and derive the corresponding Baxter equation.
Paper Structure (43 sections, 21 theorems, 481 equations, 3 figures)

This paper contains 43 sections, 21 theorems, 481 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $GL$ Toda chain
  3. Commuting Hamiltonians
  4. Eigenfunctions
  5. Baxter operator
  6. $BC$ Toda chain
  7. Commuting Hamiltonians
  8. Eigenfunctions
  9. Baxter operator
  10. Bounds and function spaces
  11. Relation to XXX spin chain
  12. Further results
  13. Reflection operator
  14. Reducing matrix equation.
  15. Solving key relations.
  16. ...and 28 more sections

Key Result

Theorem 1

The joint eigenfunctions $\Psi_{\bm{\lambda}_n}(\bm{x}_n)$ of the commuting Hamiltonians H-bc-s admit the recursive integral representation with the recursion starting from the one-particle formula Psi-1.

Figures (3)

  • Figure 1: Hankel contour and branch cut
  • Figure 2: Graph of $e^{-e^y}$
  • Figure 3: Graphs of $(1 + e^{-y})^{- g} \, (1 - e^{-y})^{g - 1}$

Theorems & Definitions (48)

  • Remark 1
  • Theorem 1
  • Example 1
  • Example 2
  • Remark 2
  • Example 3
  • Remark 3
  • Theorem 2
  • Corollary 1
  • Remark 4
  • ...and 38 more