The results exhibited in this thesis are related to Schrodinger operators in three dimensions and are subdivided in two parts based on two published papers, [15] and [14]. A variant of Turbiner's conjecture is proved in the first paper while a partial classification of quasi-exactly solvable Lie algebras of first order differential operators in dimension three is exhibited in the second paper. This classification is then used to construct new quasi-exactly solvable Schrodinger operators in three dimensions. / Turbiner's conjecture posits that, for a Lie algebraic Schrodinger operator in dimension two, the Schrodinger equation is separable if the underlying metric is locally flat. This conjecture is false in general. However, if the generating Lie algebra is imprimitive and if a certain compactness requirement holds, Rob Milson proved that in two dimensions, the Schrodinger equation separates in a Cartesian or polar coordinate system. In [15], the first paper included in this thesis, a similar theorem is proved in three variables. The imprimitivity and compactness hypotheses are still necessary and another condition, related to the underlying metric, must be imposed. In three dimensions, the separation is only partial and the separation will occur in either a spherical, cylindrical or Cartesian coordinate system. / In the second paper [14], a partial classification of quasi-exactly solvable Lie algebras of first order differential operators is performed in three dimensions. Such a classification was known in one and two dimensions but the three dimensional case was still open before the beginning of this research. These new quasi-exactly solvable Lie algebras are used to construct new quasi-exactly solvable Schrodinger operators with the property that part of their spectrum can be explicitly determined. This classification is based on a classification of Lie algebras of vector fields in three variables due to Lie and Amaldi.

No description available.

No description available.

No description available.

No description available.

One can give various rigorous definitions to the notion of "functional calculus", but a functional calculus is ultimately just a mathematically meaningful way of talking about an operator f(T), where, T is an operator and f is a function. This thesis is concerned with this concept and with one of its applications, the finding of bounds for powers of operators. It is actually this very application that has prompted the entire investigation presented here. This application is relevant to various fields, such as the numerical analysis of PDE and Markov chains. Chapter I presents various abstract approaches to the notion of "functional calculus" that are given content by three major examples: the Riesz-Dunford functional calculus, the Weyl functional calculus and the functional calculus for sectorial operators. Chapter II investigates various conditions that ensure power boundedness for operators, putting the Tadmor-Ritt condition at its center. The Riesz-Dunford calculus is instrumental for the proofs in this chapter. Chapter III investigates Pascale Vitse's use of Cauchy-Stieltjes integrals and their multipliers for obtaining bounds on powers of operators; the chapter closes with an investigation of partially power bounded operators.

Perturbations of the Laplacian are known as Schrodinger operators. We pose a question about perturbations of eigenfunctions from multiple eigenspaces and answer the question in the case of the two-sphere. This is used to extend a previously known result about nodal sets of spherical harmonics to eigenfunctions of Schrodinger operators on the two-sphere. We also review some of the classical and recent results about nodal sets and domains for eigenfunctions of the Laplacian.

Chemnitz, Techn. Univ., Diplomarb., 2006.

University, Diss., 1998--Bielefeld.

Thesis (Ph. D.)--University of North Texas, May, 2008. / Title from title page display. Includes bibliographical references.