An Inverse Spectral Problem for a Special Class of Band Matrices and Bogoyavlensky Lattice
Andrey S. Osipov · Russian Journal of Mathematical Physics · 2023
We study the inverse spectral problem method for one class of band matrices which can be applied to the integration (via Lax formalism) of Bogoyavlensky lattice $$\overset{\cdot}{a}_i=a_i^2(a_{i+1}-a_{i-1}),$$ in the finite case. Such matrices have a special structure which differs from the three-diagonal one of the matrices which appear in the Lax representation for Toda and Volterra lattices. The latter fact implies that the approach based upon the Shohat–Favard theorem and orthogonal polynomials is not applicable in this situation, as well as the use of the known continued fraction algorithms. However, it can be shown that the inverse spectral problem which amounts to reconstruction of the matrix from the moment sequence of its Weyl function, can be extended to our case. Also, the integration procedure for the finite non-Abelian Bogoyavlensky lattice by using this inverse problem method is offered.