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On Spectral Inequalities in Quantum Mechanics and Conformal Field Theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2015 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Spektralolikheter inom Kvantmekanik och Konform Fältteori (Swedish)
Abstract [en]

Following Exner et al. (Commun. Math. Phys. 26 (2014), no. 2, 531–541), we prove new Lieb-Thirring inequalities for a general class of self-adjoint, second order differential operators with matrix-valued potentials, acting in one space-dimension. This class contains, but is not restricted to, the magnetic and non-magnetic Schrödinger operators. We consider the three cases of functions defined on all reals, all positive reals, and an interval, respectively, and acquire three different kinds of bounds.

We also investigate the spectral properties of a family of operators from conformal field theory, by proving an asymptotic phase-space bound on the eigenvalue counting function and establishing a number of spectral inequalities. These bound the Riesz-means of eigenvalues for these operators, together with each individual eigenvalue, and are applied to a few physically interesting examples.

Abstract [sv]

Vi följer Exner et al. (Commun. Math. Phys. 26 (2014), nr. 2, 531–541) och bevisar nya Lieb-Thirring-olikheter för generella, andra gradens självadjungerade differentialoperatorer med matrisvärda potentialfunktioner, verkandes i en rumsdimension. Dessa innefattar och generaliserar de magnetiska och icke-magnetiska Schrödingeroperatorerna. Vi betraktar tre olika fall, med funktioner definierade på hela reella axeln, på den positiva reella axeln, samt på ett interval. Detta resulterar i tre sorters olikheter.

 Vidare undersöker vi spektralegenskaperna för en klass operatorer från konform fältteori, genom att asymptotiskt begränsa antalet egenvärden med ett fasrymdsuttryck, samt genom att bevisa ett antal spektralolikheter. Dessa begränsar Riesz-medelvärdena för operatorerna, samt varje enskilt egenvärde, och tillämpas på ett par fysikaliskt intressanta exempel.

Place, publisher, year, edition, pages
2015.
Series
TRITA-MAT-E, 2015:17
Keyword [en]
Schrödinger operators, Lieb-Thirring inequalities, commutation method, conformal field theory, Birman-Schwinger principle.
Keyword [sv]
Schrödingeroperatorer, Lieb-Thirring-olikheter, kommutationsmetoden, konform fältteori, Birman-Schwinger-principen.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-167969OAI: oai:DiVA.org:kth-167969DiVA: diva2:818366
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2015-06-08 Created: 2015-05-23 Last updated: 2015-06-18Bibliographically approved

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