| This dissertation discusses recent advances in locating the absolutely continuous spectrum of Jacobi operators by proving the boundedness of the generalized eigenfunctions of these operators. In particular, we prove two new results. One is about the discrete Schrodinger operator which is a special case of the Jacobi operator with off diagonal terms equal to 1. We show that when a potential bn of a discrete Schrodinger operator, defined on ℓ2 Z+ , slowly oscillates satisfying the conditions bn ∈ ℓ infinity and ∂bn = b n+1 - bn ∈ ℓ p, p < 2, then all solutions of the equation Ju = Eu are bounded near infinity at almost every E ∈ [-2+ limsupn→infinity bn, 2+ limsupn→infinity bn]∩[-2+ liminfn→infinity bn, 2+ liminfn→infinity bn]. We derive an asymptotic formula for generalized eigenfunctions in this case. As a corollary, by the principle of subordinacy, the absolutely continuous spectrum of the corresponding Jacobi operator is essentially supported on the same interval of E. The other result we present is about Jacobi operators with slowly decaying coefficients. We show that when the coefficients an and bn of a Jacobi operator satisfy decaying conditions an - 1, bn ∈ ℓ p, p < 2, then the solutions of the equation Ju = Eu are bounded near infinity at almost every E ∈ [-2, 2]. This produces a corollary, also by the principle of subordinacy, that the absolutely continuous spectrum of the corresponding Jacobi operator is essentially supported on the interval [-2,2]. These two results are contained in Chapter 3 and Chapter 4. |
