This thesis discusses results in the area of spectral theory of Schr"odinger operators, and their discrete analogs, Jacobi matrices, as well as some closely associated problems.

The first result we present relates to the quantum dynamics generated by a particular class of almost periodic Schr"odinger operators. We show that the dynamics generated by Schr"odinger operators whose potentials are approximated exponentially quickly by a periodic sequence exhibit a strong form of ballistic transport.

The second result exploits the connection between the KdV hierarchy and one-dimensional Schr"odinger operators to prove a uniqueness result for the KdV hierarchy with reflectionless initial data via inverse spectral theoretic techniques.

The third and fourth results concern orthogonal and Chebyshev rational functions with poles on the extended real line. In the process of extending some of the existing theory for polynomials and exploring some of the new phenomena that arise, we present a proof of a conjecture of Barry Simon's.

This thesis contains joint work with Benjamin Eichinger and Milivoje Luki'c.