van der Laan, Lars (2019) Rigged Hilbert Space Theory for Hermitian and QuasiHermitian Observables. Bachelor's Thesis, Mathematics.

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Abstract
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasihermitian 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(GelfandMaurin) 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 nonhermitian operators of interest in Quantum Mechanics, specifically PTsymmetric Hamiltonians. It is known that PTsymmetric Hamiltonians are necessarily pseudohermitian. We restrict our attention to a subclass of pseudohermitian operators, the quasihermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasihermitian 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 quasihermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasihermitian operators with respect to an operatorvalued measure exists. This is an extension of the spectral decomposition of selfadjoint operators with respect to a projectionvalued measure.
Item Type:  Thesis (Bachelor's Thesis)  

Supervisor: 


Degree programme:  Mathematics  
Thesis type:  Bachelor's Thesis  
Language:  English  
Date Deposited:  06 Jul 2019  
Last Modified:  09 Jul 2019 12:54  
URI:  http://fse.studenttheses.ub.rug.nl/id/eprint/19934 
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