Integrable Systems

5The Spectrum of Jacobi Matrices and Algebraic Curves

In this chapter, we present a study on the spectrum of periodic Jacobi matrices, infinite continued fractions, difference operators, Cauchy–Stieltjes transforms and Abelian integrals.

5.1. Jacobi matrices and algebraic curves

A Jacobi matrix is a doubly infinite matrix (a_ij) for i, j ∈ ℤ such that: a_ij = 0 if |i − j| is large enough. The set of these matrices forms an associative algebra and consequently a Lie algebra by antisymmetrization. Consider the Jacobi matrix

[5.1]

As an example of V_{−j, k} (theorem 4.4), consider the infinite Jacobi matrix (symmetric, tridiagonal and N-periodic):

[5.2]

with a_i, b_i ∈ ℂ. The matrix A is N-periodic when a_i+N = a_i, b_i+N = b_i, ∀i ∈ ℤ. We denote by f = (..., f₋₁, f₀, f₁, ...) the (infinite) column vector and by D the operator passage of degree +1, Df_i = f_i+1. Since the matrix A is N-periodic, we have AD^N = D^N A. Reciprocally, this relation of commutation means that N is the period of A. Let

be the finite Jacobi matrix (symmetric tridiagonal and N-periodic). The determinant of the matrix

is