Beijing Institute of Mathematical Sciences and Applications

A quantum Toda chain is a quantum system of n interacting particles living on a line.

In many respects, this model is a more complicated analogue of two simple but nontrivial quantum systems — the system of free particles and the harmonic oscillator.
All these models can be solved exactly, in the sense that it is possible
to construct an orthogonal and complete set of eigenfunctions for the Hamiltonian.

The quantum Toda chain is an example of a quantum integrable system and can be used to clearly demonstrate the methods developed for studying these systems.

The introductory part of the course.

The simplest quantum mechanical systems

1) Free particle
In any case the solution of this problem is a starting point for the solution of n-particle problem for the Toda chain.
2) Harmonic oscillator
Schroedinger representation
Fock representation
Creation and annihilation operators
The unitary equivalence of representations

Sidney Coleman once said "The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction."

This is true in the present case and construction of the eigenfunction in the case of open Toda chain is highly nontrivial generalization of the construction of eigenfunctios of harmonic oscillator.

In the first part of the course we will consider the following topics

1) Quantum Toda open chain. Hamiltonian and family of commuting operators.
2) Lax operator, monodromy matrix and Yang-Baxter commutation relations.
3) Iterative construction of common eigenfunctions of the family of commuting operators.
4) Ladder-operators. Iterative construction of eigenfunction in coordinate space -- Gauss-Givental representation.
5) Iterative construction of eigenfunction in spectral-space -- Mellin-Barnes representation.
6) Calculation of the scalar product: Orthogonality and the Sklyanin measure. Feynman diagrams. Proof of the main relations.
7) Equivalence of the Gauss-Givental and the Mellin-Barnes representations.
8) Gustafson integrals and completeness.

In the first part we will concentrate on the exact integral representations for the eigenfunctions and the proofs of the main integral identities. The maing goal is to demonstrate in details how everything works and to show the main technical tricks.

In the second part of the course we will concentrate on the operator solution of the Yang-Baxter equation and the role of these integral R-operators in construction of eigenfunctions.

We are going to consider the following topics.
1) Solution of the Yang-Baxter equation and integral R-operators.
2) Q-operator. Definition, kernel identity. Baxter equation.
3) Sklyanin representation of separated variables. Periodic Toda chain.
4) Inverse problem -- solution by O.Babelon and E.Sklyanin.
5) Quasiclassical approximation. Classical Baecklund transformation. Solution of the classical open Toda chain. Solution of the scattering problem for the classical open Toda chain.
6) Reflection equations. BC-Toda chain. Construction of eigenfunctions.