In this thesis we characterize the spectrum of one-dimensional Schrödinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of Schrödinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the Schrödinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.
Construction and Qualitative Behavior of the Solution of the Perturbated Riemann Problem for the System of One-Dimensional Isentropic Flow with Damping
