Numerical analysis of the spectra of dissipative Schrodinger-type and related operators

Spectral problems of band-gap structure appear in various applications such as elasticity theory, electromagnetic waves, and photonic crystals. In the numerical approximation of these problems an important phenomenon known as spectral pollution arises due to the discretisation process. In this thesis we focus on two different techniques to calculate eigenvalues in spectral gaps of Schr¨odinger-type operators which are free of spectral pollution. The original material in this thesis is based on papers [7], [8], and [6]. The material in these papers is explained in details in Chapter 4, Chapter 5, and Chapter 6 with summaries presented in Chapter 2 and Chapter 3, respectively. In Chapter 4, we investigate approximation of eigenvalues in spectral gaps of Schrodinger operators with matrix coefficients. We employ the dissipative barrier technique and domain truncation and analyse spectral properties of the resulting operators. Our theoretical foundations are based on the notions of Floquet theory and Dirichlet-to-Neumann maps. The effectiveness of this technique is illustrated through different numerical examples including a model in optics. In Chapter 5, we study approximation of isolated eigenvalues in spectral gaps of elliptic partial differential operators for models of semi-infinite waveguides. The appproximation is obtained using the interaction of the dissipative technique and domain truncation of the operators. Our theoretical results are based on the error estimate of the Dirichlet-to-Neumann maps on the cross-section of the waveguides and perturbation determinants. Some numerical examples on waveguides are indicated to show the effectiveness of the presented technique. In Chapter 6, we propose a numerical algorithm to calculate eigenvalues of the perturbed periodic matrix-valued Schrodinger operators which are located in spectral gaps. The spectral-pollution-free algorithm is based on combining shooting with Floquet theory, as well as Atkinson Θ−matrices, to avoid the associated stiffness problems and allow eigenvalue counting. We derive interesting new oscillation results. As far as we know these are the first oscillation theory results for matrix Schrodinger operators for λ in a spectral gap above the first spectral band. Numerical examples show that this method gives more accurate results and requires less time than those obtained from the finite difference methods, which are coupled with contour integral λ−nonlinear eigenvalue problems. In addition, the proposed method gives better results than the dissipative barrier scheme with domain truncation which lead to λ−linear eigenvalue problems