Spectral asymptotics associated with Dirac-type operators

dc.contributor.advisor Sukochev, Fedor en_US Vella, Dominic en_US 2022-03-23T12:25:36Z 2022-03-23T12:25:36Z 2019 en_US
dc.description.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 $\zeta$-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. en_US
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri en_US
dc.title Spectral asymptotics associated with Dirac-type operators en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Vella, Dominic
dspace.entity.type Publication en_US
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Vella, Dominic, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Sukochev, Fedor, Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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