Matrix Model and Stationary Problem in Toda Chain

TL;DR

This paper explores the stationary problem in the Toda chain, revealing its connection to multi-support solutions of matrix models, and provides explicit calculations for Hamiltonians and symplectic forms, along with quantum problem insights.

Contribution

It establishes a precise link between Toda chain stationary solutions and multi-support matrix model solutions, including explicit Hamiltonian and symplectic form calculations.

Findings

Geometric data correspond to multi-support solutions of matrix models.

Explicit Hamiltonian and symplectic form calculations for initial examples.

Discussion of quantum problem properties and future perspectives.

Abstract

We analyze the stationary problem for the Toda chain, and show that arising geometric data exactly correspond to the multi-support solutions of one-matrix model with a polynomial potential. For the first nontrivial examples the Hamiltonians and symplectic forms are calculated explicitly, and the consistency checks are performed. The corresponding quantum problem is formulated and some its properties and perspectives are discussed.