On the finiteness of logarithmic Hamiltonians for Volterra-type lattices in terms of the spectral measures of Jacobi operators

TL;DR

This work investigates when the logarithmic Hamiltonian $H_0=\frac{1}{2}\sum_n \ln a_n$ of Volterra-type lattices remains finite and ties this finiteness to the spectral measures of associated Jacobi operators. Employing the inverse spectral theory, Nikishin’s orthogonal-polynomial framework, Szegő-type entropy conditions, and Lax-pair representations, it establishes a precise correspondence between finite $H_0$ and even spectral measures with essential support in $[-2,2]$ satisfying the Szegő condition, with an extension to the infinite lattice via a spectral-matrix formalism and the Peherstorfer–Yuditskii criterion for infinite point spectrum. The analysis also covers the modified Volterra lattice and shows how measure evolution $d\rho(\lambda,t)=K(t)e^{\lambda^2 t} d\rho(\lambda,0)$ yields a time-parametrized family of finite-$H_0$ lattices, while ensuring the essential spectral support and mass-point structure are controlled. This provides a rigorous bridge between integrable lattice dynamics and classical spectral theory of Jacobi operators, with potential implications for inverse problems and extensions to related Bogoyavlensky-type systems.

Abstract

We establish a correspondence between the semi-infinite and infinite Volterra lattices having a finite logarithmic Hamiltonian and certain classes of even probability measures. In doing so, we apply the inverse spectral theory of Jacobi operators and the theory of orthogonal polynomials. A similar correspondence is established for semi-infinite modified Volterra lattices.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Corollary 2
• Theorem 4
• Theorem 5
• Theorem 6