## Publication: Spectral asymptotics associated with Dirac-type operators

 dc.contributor.advisor Sukochev, Fedor en_US dc.contributor.author Vella, Dominic en_US dc.date.accessioned 2022-03-23T12:25:36Z dc.date.available 2022-03-23T12:25:36Z dc.date.issued 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.identifier.uri http://hdl.handle.net/1959.4/66118 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 https://creativecommons.org/licenses/by-nc-nd/3.0/au/ 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.accessRights.uri https://purl.org/coar/access_right/c_abf2 unsw.identifier.doi https://doi.org/10.26190/unsworks/21792 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 unsw.relation.school School of Mathematics & Statistics * unsw.thesis.degreetype PhD Doctorate en_US
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