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Construction of solutions of Toda lattices by the classical moment problem

Alexander Mikhaylov, Victor Mikhaylov

TL;DR

This work develops a constructive framework for solving the semi‑infinite Toda lattice with broad (even unbounded) initial data by recasting the problem in terms of the classical moment problem for Jacobi operators. It derives a Moser‑type evolution for the moments $s_k(t)$ of the spectral measure under the Toda flow, linking the dynamics to the spectral data and enabling finite and infinite‑dimensional analyses via de Branges spaces. By establishing positivity criteria for Hankel matrices and leveraging moment–inversion formulas, the authors show how to recover time‑dependent Jacobi coefficients $(a_n(t), b_n(t))$ from the evolving moments, providing a rigorous route to construct Toda solutions beyond bounded data. The approach unifies spectral theory, moment problem theory, and boundary control methods, and yields explicit recurrence relations for the moments that define the evolving Jacobi operator and hence the Toda solution. This extends the applicability of Toda dynamics to a wider class of initial conditions and offers a new perspective on the interplay between moment problems and integrable systems.

Abstract

Making use of formulas of J. Moser for a finite-dimensional Toda lattices, we derive the evolution law for moments of the spectral measure of the semi-infinite Jacobi operator associated with the Toda lattice. This allows us to construct solutions of semi-infinite Toda lattices for a wide class of unbounded initial data by using well-known results from the classical moment problem theory.

Construction of solutions of Toda lattices by the classical moment problem

TL;DR

This work develops a constructive framework for solving the semi‑infinite Toda lattice with broad (even unbounded) initial data by recasting the problem in terms of the classical moment problem for Jacobi operators. It derives a Moser‑type evolution for the moments of the spectral measure under the Toda flow, linking the dynamics to the spectral data and enabling finite and infinite‑dimensional analyses via de Branges spaces. By establishing positivity criteria for Hankel matrices and leveraging moment–inversion formulas, the authors show how to recover time‑dependent Jacobi coefficients from the evolving moments, providing a rigorous route to construct Toda solutions beyond bounded data. The approach unifies spectral theory, moment problem theory, and boundary control methods, and yields explicit recurrence relations for the moments that define the evolving Jacobi operator and hence the Toda solution. This extends the applicability of Toda dynamics to a wider class of initial conditions and offers a new perspective on the interplay between moment problems and integrable systems.

Abstract

Making use of formulas of J. Moser for a finite-dimensional Toda lattices, we derive the evolution law for moments of the spectral measure of the semi-infinite Jacobi operator associated with the Toda lattice. This allows us to construct solutions of semi-infinite Toda lattices for a wide class of unbounded initial data by using well-known results from the classical moment problem theory.
Paper Structure (6 sections, 9 theorems, 75 equations)

This paper contains 6 sections, 9 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Finite Toda lattice. Moser formula.
  3. Moments of the spectral measure of finite and semi-infinite Jacobi operators and de Branges spaces.
  4. Evolution of moments.
  5. Finite-dimensional case
  6. More on finite-dimensional case, infinite-dimensional case.

Key Result

Proposition 1

The eigenvalues of the matrix $H_N(t)$ do not depend on $t$: $\lambda_{N,\,j}(t)=\lambda_{N,\,j}(0)=\lambda_{N,\,j}$.

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Theorem 1
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • ...and 2 more