Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables

Abstract

In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.