Oscillation Theory and Renormalized Oscillation Theory for Jacobi Operators

We provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: If u solves the Jacobi equation (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n) = λu(n), λ ∈ R (in the weak sense) on an arbitrary interval and satisfies the boundary condition on the left or right, then the dimension of the spectral projection P(−∞,λ)(H) of H equals the number of nodes (i.e., sign flips if a(n) < 0) of u. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory; if λ1,2 ∈ R and if u1,2 solve the Jacobi equation Huj = λjuj , j = 1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection P(λ1,λ2)(H) equals the number of nodes of the Wronskian of u1 and u2. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators.