Abstract :
We provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: Ifusolves the Jacobi equation (Hu)(n)=a(n)u(n+1)+a(n−1)u(n−1)−b(n)u(n)=λu(n),λ (in the weak sense) on an arbitrary interval and satisfies the boundary condition on the left or right, then the dimension of the spectral projectionP(−∞, λ)(H) ofHequals the number of nodes (i.e., sign flips ifa(n)<0) ofu. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory; ifλ1, 2 and ifu1, 2solve the Jacobi equationHuj=λjuj,j=1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projectionP(λ1, λ2)(H) equals the number of nodes of the Wronskian ofu1andu2. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators.
