The spectrum of Jacobi matrices

Abstract

Let L be a periodic symmetric tridiagonal matrix of size N; "periodic" means that L has one extra-entry in the upper right corner and by symmetry in the lower left one. Let b i be the diagonal and ai the subdiagonal entries. The present paper deals with the space ~ ' of such matrices with a given spectrum. On Jr' there is a natural class of commuting flows (isospectral deformations), which derive from Hamiltonian mechanics. When the given spectrum is non-degenerate, there are N 1 independent flows except for some degeneracies on some lower dimensional submanifolds. Each of these flows has in general N 1 integrals in involution, so that generically the solutions are quasi-periodic, their orbits are dense on a N 1 dimensional torus and there exists a canonical transformation to a set of action-angle variables. However there is much more involved, because these tori are algebraic surfaces and their periods can be expressed in terms of hyperelliptic functions; this is to say each such torus is a Jacobi variety. The transformation from the original variables (a~, bi) to a set of separation variables (/~i, vi) is of rational character. The "posi t ion" components p~ of these variables are provided by the spectrum of the matrix L, from which the first row and the first column has been removed. They define a local system of coordinates on the torus. Another system of coordinates t~ is provided by the group action of R N1 on the torus, such that the flows appear as linear motions on the torus. The Jacobi transformation maps the local system of coordinates (#1 . . . . . #N-l) into the global one (q . . . . . tN_l). The inverse map can be expressed in terms of the flows above and can be explicited in terms of quotients of theta functions invoking the theory of the Jacobi inversion problem. As a bonus, this yields explicit solutions to the differential equation defined by the isospectral flows, in terms of Abelian and theta functions. Finally, ~t' can be foliated by N-1-d imens iona l tori, each of which can be labelled by a modulus; this modulus is defined as the product of the non-diagonal