Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53).

Not available / text

The purpose of this dissertation is to review the salient features of non-Hermitian quantum mechanics. An introduction to Hermitian quantum mechanics is included to make this review as accessible as possible. Attempts at formulating a consistent physical theory are introduced, before examining non-Hermitian theories' uses as convenient computational frameworks. Particular emphasis is placed on recent developments in open quantum systems that utilise non-Hermitian Hamiltonians. Chapter four introduces a logic that maps a non-Hermitian Hamiltonian onto a non-Hamiltonian algebra that has a Hermitian Hamiltonian. This was put forward by Sergi, who then goes on to show its application to a two level system. The time evolution is then derived in terms of the density matrix model. This system can then be used to analyse di erent types of decay such as coherence and population di erence. This serves to illustrate the usefulness of the approach. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph.D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53) Also available in print.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Also available online.

Thesis (M.S.M.E.)--University of Massachusetts Amherst, 2008. / Includes bibliographical references (p. 68-71).

In this thesis, we study three problems in complex geometry. In the first part, we study the behavior of the Kahler-Ricci flow on complete non-compact manifolds with negative holomorphic curvature. We show that Kahler-Ricci flow converges to a Kahler-Einstein metric when the initial manifold admits a suitable exhaustion function, thus improving upon a result of D. Wu and S.T. Yau. These results are partly obtained in joint work with S. Huang, M.-C. Lee and L.-F. Tam. In the second part of this thesis, we introduce a new Kodaira-Bochner type formula for closed (1, 1)-form in non-Kahler geometry. Based on this new formula, We propose a new curvature positivity condition in non-Kahler manifolds and proved a strong rigidity type theorem for manifolds satisfying this curvature positivity condition. We also find interesting examples non-Kahler manifolds satisfying the curvature positivity condition in a class of manifolds called Vaisman manifolds. In the third part of this thesis, we study the degenerations of asymptotically conical Calabi-Yau manifolds as the Kahler class degenerates to a non-Kahler class. Under suitable hypothesis, we prove the convergence of asymptotically conical Calabi-Yau metrics to a singular asymptotically conical Calabi-Yau current with compactly supported singularities. Using this, we construct singular asymptotically conical Calabi-Yau metrics on non-compact singular varieties and identify the topology of these singular metrics with the singular variety. We also give some interpretations of these asymptotically conical Calabi-Yau metrics from the point of view of physics. These results are obtained in joint work with T. Collins and B. Guo.

I den här avhandlingen introducerar vi ämnen som är centrala för de bifogade artiklarna [Art. I, Art. II, Art. III]. Alla tre handlar om degenererade faser i icke-hermitska system. I kapitel 1 inleder vi med att introducera experimentella plattformar för icke-hermitska system, och skillnaderna i deras matematiska hantering gentemot hermitska system. I [Art. I] undersökte vi tvådimensionella icke-hermitska system under periodiska randvillkor, och fann att nielsen-ninomiyas teorem inte kan tillämpas trivialt i detta fall. Vi visar hur detta teorem fungerar för hermitska system i kapitel 2, och förklarar varför det misslyckas för icke-hermitska system. I detta kapitel förklaras även den allmänna metoden för homotopiklassificering utifrån begreppet degenererad hamiltonian. Slutligen diskuterar vi diskreta symmetrier i kapitel 3, med utgångspunkt i kristallina symmetrier. Dessa är centrala för [Art. II], där vi visade att en hamiltonian som beskriver kristaller med specifika symmetri-grupper måste ha degenererade punkter. Fortsättningsvis diskuterar vi anti-unitära och partikel-hål-symmetrier, vilket leder till den tiofaldiga klassificeringen av symmetrier. Vi betonar betydelsen av tidsomvändningssymmetri och PT-symmetri, en kombination av tidsomvändning och rumslig inversion. Denna symmetri förstärker ytterligare den degenererade struktur som vi fann i [Art. II], vilket leder till exceptionella linjer, ett fenomen som är omöjligt i hermitska system. PT-symmetrin är också central för [Art. III], där vi klassificerade homotopistrukturen för PT-symmetriska system i allmänhet och fann ett antal intressanta topologiska invarianter.