The present thesis discusses various aspects of spectral theory for Jacobi operators.
The first chapter reviews Weyl-Titchmarsh theory for these operators and provides the necessary background for the following chapters.
In the second chapter we provide a comprehensive treatment of oscillation the- ory for Jacobi operators with separated boundary conditions. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators.
In the third chapter we offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded back- ground Jacobi operator and the double commutation method which inserts eigen- values into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace cor- responding to the newly inserted eigenvalues, with the original background operator. Finally, we show how to iterate the above methods. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and the N-soliton solutions of the Toda and Kac-van Moerbeke lattice equations with respect to arbitrary background solutions.