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

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.