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Resonances of Dirac Operators
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2014.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 84
Keyword [en]
Semiclassical analysis, Dirac operator, resonances, Poisson trace formula, scattering theory, complex absorbing potential, pseudodifferential operator
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-223841ISBN: 978-91-506-2400-7 (print)OAI: oai:DiVA.org:uu-223841DiVA: diva2:714342
Public defence
2014-06-09, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, plan 0, Hus 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2014-05-21 Created: 2014-04-27 Last updated: 2014-05-21
List of papers
1. Complex absorbing potential method for the perturbed Dirac operator
Open this publication in new window or tab >>Complex absorbing potential method for the perturbed Dirac operator
2014 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 39, no 8, 1451-1478 p.Article in journal (Refereed) Published
Abstract [en]

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for nonrelativis- tic and relativistic Hamiltonians. In the semiclassical limit h → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudod- ifferential operator theory and microlocal analysis.

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-223457 (URN)10.1080/03605302.2014.908910 (DOI)000340490700003 ()
Available from: 2014-04-20 Created: 2014-04-20 Last updated: 2017-12-05Bibliographically approved
2. Complex absorbing potential method for Dirac operators: clusters of resoances
Open this publication in new window or tab >>Complex absorbing potential method for Dirac operators: clusters of resoances
2014 (English)In: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 71, no 1, 259-283 p.Article in journal (Refereed) Published
Abstract [en]

For both nonrelativistic and relativistic Hamiltonians, the complex absorbing potential (CAP) method has been applied extensively to cal- culate resonances in physics and chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-223458 (URN)10.7900/jot.2012feb13.1948 (DOI)000334260100011 ()
Available from: 2014-04-20 Created: 2014-04-20 Last updated: 2017-12-05Bibliographically approved
3. Existence of Dirac resonances in the semi-classical limit
Open this publication in new window or tab >>Existence of Dirac resonances in the semi-classical limit
2014 (English)In: Dynamics of Partial Differential Equations, ISSN 1548-159X, E-ISSN 2163-7873, Vol. 11, no 4, 381-395 p.Article in journal (Refereed) Published
Abstract [en]

We study the existence of quantum resonances of the three-dimensional semiclassical Dirac operator perturbed by smooth, bounded and real-valued scalar potentials V decaying like < x >(-delta) at infinity for some delta > 0. By studying analytic singularities of a certain distribution related to V and by combining two trace formulas, we prove that the perturbed Dirac operators possess resonances near sup V + 1 and inf V - 1. We also provide a lower bound for the number of resonances near these points expressed in terms of the semiclassical parameter.

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-223459 (URN)000348668600005 ()
Available from: 2014-04-20 Created: 2014-04-20 Last updated: 2017-12-05Bibliographically approved
4. Poisson wave trace formula for perturbed Dirac operators
Open this publication in new window or tab >>Poisson wave trace formula for perturbed Dirac operators
2017 (English)In: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 77, no 1, 133-147 p.Article in journal (Refereed) Published
Abstract [en]

We consider self-adjoint Dirac operators D = D-0 + V(x), where D 0 is the free three-dimensional Dirac operator and V(x) is a smooth compactly supported Hermitian matrix. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of D.

Keyword
Dirac operator, resonances, Poisson wave trace
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-223460 (URN)10.7900/jot.2016mar04.2119 (DOI)000396724500008 ()
Available from: 2014-04-20 Created: 2014-04-20 Last updated: 2017-12-05Bibliographically approved

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