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Spectral Asymptotics Associated with Dirac-Type Operators

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Spectral Asymptotics Associated with Dirac-Type Operators Spectral asymptotics associated with Dirac-type operators Dominic Michael Vella A thesis in fulfulment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics Faculty of Science August 2019 Thesis/Dissertation Sheet Surname/Family Name : Vella Given Name/s : Dominic Michael Abbreviation for degree as given in the University calendar : PhD Faculty : Science School : Mathematics and Statistics Thesis Title : Spectral asymptotics associated with Dirac-type operators Abstract 350 words maximum: (PLEASE TYPE) This thesis is concerned first with a non-compact variation of Connes' trace theorem, which demonstrated that the Dixmier trace extends the notion of Lebesgue integration on a compact manifold. To obtain the variation, we develop a new $\zeta$-residue formula, which is proved by an innovative approach using double operator integrals. Using this formula, Connes' trace theorem is shown for operators of the form $M_f(1-\Delta)^{- \frac{d}{2}}$ on $L_2(\mathbb{R}^d)$, where $M_f$ is multiplication by a function belonging to the Sobolev space $W_1^d(\mathbb{R}^d)$---the space of all integrable functions on $\mathbb{R}^d$ whose weak derivatives up to order $d$ are all also integrable---and $\Delta$ is the Laplacian on $L_2(\mathbb{R}^d)$. An analogous formula for the Moyal plane is also shown. The $\zeta$-residue formula we derive also enables a second result. We consider the smoothed Riesz map $\mathrm{g}$ of the massless Dirac operator $\mathcal{D}$ on $\mathbb{R}^d$, for $d\geq 2$, and study its properties in terms of weak Schatten classes. Our sharp estimates, which are optimal in the scale of weak Schatten classes, show that the decay of singular values of $\mathrm{g}(\mathcal{D}+V)-\mathrm{g}(\mathcal{D})$ differs dramatically for the case when the perturbation $V$ is a purely electric potential and the case when $V$ is a magnetic one. The application of double operator integrals also yields a similar result for the operator $f(\mathcal{D}+V)-f(\mathcal{D})$ for an arbitrary monotone function $f$ on $\mathbb{R}$ whose derivative is Schwartz. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). …………………………………………………………… ……………………………………..……………… ……….……………………...…….… Signature Witness Signature Date The University recognises that there may be exceptional circumstances requiring restrictions on copying or conditions on use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of restriction may be considered in exceptional circumstances and require the approval of the Dean of Graduate Research. FOR OFFICE USE ONLY Date of completion of requirements for Award: ORIGINALITY STATEMENT ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’ Signed …………………………………………….............. Date …………………………………………….............. INCLUSION OF PUBLICATIONS STATEMENT UNSW is supportive of candidates publishing their research results during their candidature as detailed in the UNSW Thesis Examination Procedure. Publications can be used in their thesis in lieu of a Chapter if: • The student contributed greater than 50% of the content in the publication and is the “primary author”, i.e. the student was responsible primarily for the planning, execution and preparation of the work for publication • The student has approval to include the publication in their thesis in lieu of a Chapter from their supervisor and Postgraduate Coordinator. • The publication is not subject to any obligations or contractual agreements with a third party that would constrain its inclusion in the thesis Please indicate whether this thesis contains published material or not. This thesis contains no publications, either published or submitted for publication ☐ Some of the work described in this thesis has been published and it has been ☒ documented in the relevant Chapters with acknowledgement This thesis has publications (either published or submitted for publication) ☐ incorporated into it in lieu of a chapter and the details are presented below CANDIDATE’S DECLARATION I declare that: • I have complied with the Thesis Examination Procedure • where I have used a publication in lieu of a Chapter, the listed publication(s) below meet(s) the requirements to be included in the thesis. Name Signature Date (dd/mm/yy) Dominic Michael Vella v COPYRIGHT STATEMENT ‘I hereby grant the University of New South Wales or its agents a non-exclusive licence to archive and to make available (including to members of the public) my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known. I acknowledge that I retain all intellectual property rights which subsist in my thesis or dissertation, such as copyright and patent rights, subject to applicable law. I also retain the right to use all or part of my thesis or dissertation in future works (such as articles or books).’ ‘For any substantial portions of copyright material used in this thesis, written permission for use has been obtained, or the copyright material is removed from the final public version of the thesis.’ Signed ……………………………………………........................... Date …………………………………………….............................. AUTHENTICITY STATEMENT ‘I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis.’ Signed ……………………………………………........................... Date …………………………………………….............................. Abstract This thesis is concerned first with a non-compact variation of Connes' trace theorem, which demonstrated that the Dixmier trace extends the notion of Lebesgue integration on a compact manifold. To obtain the variation, we develop a new ζ-residue formula, which is proved by an innovative approach using double operator integrals. Using this formula, − d d Connes' trace theorem is shown for operators of the form Mf (1 − ∆) 2 on L2(R ), where d d Mf is multiplication by a function belonging to the Sobolev space W1 (R )|the space of all d integrable functions on R whose weak derivatives up to order d are all also integrable| d and ∆ is the Laplacian on L2(R ). An analogous formula for the Moyal plane is also shown. The ζ-residue formula we derive also enables a second result. We consider the smoothed d Riesz map g of the massless Dirac operator D on R , for d ≥ 2, and study its properties in terms of weak Schatten classes. Our sharp estimates, which are optimal in the scale of weak Schatten classes, show that the decay of singular values of g(D + V ) − g(D) differs dramatically for the case when the perturbation V is a purely electric potential and the case when V is a magnetic one. The application of double operator integrals also yields a similar result for the operator f(D + V ) − f(D) for an arbitrary monotone function f on R whose derivative is Schwartz. iii iv Acknowledgements At some point near the end of 2016, a starry-eyed student with a fresh bachelor's degree in mathematics and a mind to pursue a PhD in mathematical physics visited Alan Carey at ANU. At that meeting, Alan advised the student that the prolific academic and his close collaborator, Fedor Sukochev, was likely a good person to speak to. Though not a mathematical physicist, Fedor's research in noncommutative geometry had inextricable roots in the area. Moreover, he worked at UNSW, the university where the student had conducted his undergrad studies. Within days, Fedor found the na¨ıve stranger of a student knocking at his office door and (quite uncharacteristically, I might add) decided to accept him as a student almost immediately! I know not by what rubric he made that remarkable decision, but it was due to that decision|and Alan's direction|that the student was able to eventually produce the present thesis. Fedor's supervision was as unconventional as it was effective. His broad insights, wis- dom and perfectionism were all inspiring to behold and were a constant driver for the direction of my research, but balancing his own research projects with all of his research students, of which there were both numerous, was a nontrivial task. For this reason, I benefitted from the co-supervision of correspondingly numerous academics, each of whom were tasked with fostering a certain set of skills. Dima's door was always open, and he was constantly brimming with keen mathematical insights. He encouraged the pursuit of beauty and generality in results proven through clever argumentation. Steven countered this by ensuring my approach to mathematics remained grounded, rigorous, clearly written and timely, with his careful and time-consuming reading of my work time and time again resulting in essential course-correction.
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