Automatic computation of quantum-mechanical bound states and wavefunctions

We discuss the automatic solution of the multichannel Schr\"odinger equation. The proposed approach is based on the use of a CP method for which the step size is not restricted by the oscillations in the solution. Moreover, this CP method turns out to form a natural scheme for the integration of the Riccati differential equation which arises when introducing the (inverse) logarithmic derivative. A new Pr\"ufer type mechanism which derives all the required information from the propagation of the inverse of the log-derivative, is introduced. It improves and refines the eigenvalue shooting process and implies that the user may specify the required eigenvalue by its index

Study of special algorithms for solving Sturm-Liouville and Schrodinger equations

In dit proefschrift beschrijven we een specifieke klasse van numerieke methoden voor het oplossen van Sturm-Liouville en Schrodinger vergelijkingen. Ook de Matlab-implementatie van de methoden wordt besproken

Spectra of self-adjoint extensions and applications to solvable Schroedinger operators

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.Comment: 81 pages, new references added, subsection 1.3 extended, typos correcte

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

Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential

We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, self-adjoint in $L^2(\R^2)$. Here $A$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption $\lim_{x \to \infty}(W(x) - W(-x)) < 2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm} = H_0 \pm V$ where the electric potential $V \in L^{\infty}(\R^2)$ is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of $H_\pm$ in the spectral gaps of $H_0$. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to $V$. Further, we restrict our attention on perturbations $V$ of compact support and constant sign. We establish a geometric condition on the support of $V$ which guarantees the finiteness of the eigenvalues of $H_{\pm}$ in any spectral gap of $H_0$. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of $H_+$ (resp. $H_-$) to the left (resp. right) edge of a given spectral gap, is Gaussian.Comment: 32 page