published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

No description available.

No description available.

Much of operator theory hangs its coat on the spectral theorem, but the latter is exclusive to normal operators. Likewise, isometries are well understood via the Wold decomposition. It is von Neumann's inequality that enables a functional calculus for arbitrary contractions on Hilbert spaces. There are essentially two avenues that lead to von Neumann, one being the analytical theory of positive maps, the other marked by geometric dilation theorems. These diverse lines of approach are in fact unified by the inequality. Although our main focus is von Neumann's inequality, for which we provide four different proofs, we shall, however, periodically indulge in some of its intricate cousins.

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.