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Spectral estimates for the magnetic Schrödinger operator and the Heisenberg Laplacian
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [sv]

I denna avhandling, som omfattar fyra forskningsartiklar, betraktas två operatorer inom den matematiska fysiken.

De båda tidigare artiklarna innehåller resultat för Schrödingeroperatorn med Aharonov-Bohm-magnetfält. I artikel I beräknas spektrum och egenfunktioner till denna operator i R2 explicit i ett antal fall då en radialsymmetrisk skalärvärd potential eller ett konstant magnetfält läggs till. I flera av de studerade fallen kan den skarpa konstanten i Lieb-Thirrings olikhet beräknas för γ = 0 och γ ≥ 1.

I artikel II bevisas semiklassiska uppskattningar för moment av egenvärdena i begränsade tvådimensionella områden. Vidare presenteras ett exempel då den generaliserade diamagnetiska olikheten, framlagd som en förmodan av Erdős, Loss och Vougalter, är falsk. Numeriska studier kompletterar dessa resultat.

De båda senare artiklarna innehåller ett flertal spektrumuppskattningar för Heisenberg-Laplace-operatorn. I artikel III bevisas skarpa olikheter för spektret till Dirichletproblemet i (2n + 1)-dimensionella områden med ändligt mått.

Låt λk och μk beteckna egenvärdena till Dirichlet- respektive Neumannproblemet i ett område med ändligt mått. N. D. Filonov har bevisat olikheten μk+1 < λk för den euklidiska Laplaceoperatorn. I artikel IV visas detta resultat för Heisenberg-Laplaceoperatorn i tredimensionella områden som uppfyller vissa geometriska villkor.

Abstract [en]

In this thesis, which comprises four research papers, two operators in mathe- matical physics are considered.

The former two papers contain results for the Schrödinger operator with an Aharonov-Bohm magnetic field. In Paper I we explicitly compute the spectrum and eigenfunctions of this operator in R2 in a number of cases where a radial scalar potential and/or a constant magnetic field are superimposed. In some of the studied cases we calculate the sharp constants in the Lieb-Thirring inequality for γ = 0 and γ ≥ 1.

In Paper II we prove semi-classical estimates on moments of the eigenvalues in bounded two-dimensional domains. We moreover present an example where the generalised diamagnetic inequality, conjectured by Erdős, Loss and Vougalter, fails. Numerical studies complement these results.

The latter two papers contain several spectral estimates for the Heisenberg Laplacian. In Paper III we obtain sharp inequalities for the spectrum of the Dirichlet problem in (2n + 1)-dimensional domains of finite measure.

Let λk and μk denote the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure. N. D. Filonov has proved that the inequality μk+1 < λk holds for the Euclidean Laplacian. In Paper IV we extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfil certain geometric conditions.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , vii, 45 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08:01
Keyword [en]
spectral theory; Schrödinger operator; magnetic field; Aharonov-Bohm; Lieb-Thirring; diamagnetic inequality; Heisenberg group; Heisenberg Laplacian; hypoelliptic
Keyword [sv]
spektralteori; Schrödingeroperator; magnetfält; Aharonov-Bohm; Lieb-Thirring; diamagnetisk olikhet; Heisenberggrupp; Heisenberg-Laplace-operator; hypoelliptisk
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4578ISBN: 978-91-7178-798-9 (print)OAI: oai:DiVA.org:kth-4578DiVA: diva2:12951
Public defence
2008-01-11, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100712Available from: 2007-12-12 Created: 2007-12-12 Last updated: 2010-07-12Bibliographically approved
List of papers
1. On the spectrum and eigenfunctions of the Schrödinger operator with Aharonov-Bohm magnetic field
Open this publication in new window or tab >>On the spectrum and eigenfunctions of the Schrödinger operator with Aharonov-Bohm magnetic field
2005 (English)In: International journal of mathematics and mathematical sciences, ISSN 0161-1712, E-ISSN 1687-0425, Vol. 2005, no 23, 3751-3766 p.Article in journal (Refereed) Published
Abstract [en]

We explicitly compute the spectrum and eigenfunctions of the magnetic Schrödinger operator H (A, V) = (i∇ + A)2 + V in L2(ℝ2), with Aharonov-Bohm vector potential, A(x1,x2) = α(-x2,x1)/ x 2, and either quadratic or Coulomb scalar potential V. We also determine sharp constants in the CLR inequality, both dependent on the fractional part of α and both greater than unity. In the case of quadratic potential, it turns out that the LT inequality holds for all γ ≥ 1 with the classical constant, as expected from the nonmagnetic system (harmonic oscillator).

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7813 (URN)10.1155/IJMMS.2005.3751 (DOI)
Note
QC 20100712Available from: 2007-12-12 Created: 2007-12-12 Last updated: 2010-07-12Bibliographically approved
2. Eigenvalue estimates for the Aharonov-Bohm operator in a domain
Open this publication in new window or tab >>Eigenvalue estimates for the Aharonov-Bohm operator in a domain
2009 (English)In: Methods of spectral analysis in mathematical physics, 2009, 115-137 p.Conference paper, Published paper (Other academic)
Abstract [en]

We prove semi-classical estimates on moments of eigenvalues of the Aharonov-Bohm operator in bounded two-dimensional domains. Moreover, we present a counterexample to the generalized diamagnetic inequality which was proposed by Erdos, Loss and Vougalter. Numerical studies complement these results.

Series
Operator theory advances and applications, ISSN 0255-0156 ; 186
Keyword
Spectral estimates; Berezin-Li-Yau inequalities; Aharonov-Bohm magnetic field
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7814 (URN)000262459300007 ()978-3-7643-8754-9 (ISBN)
Conference
International Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP 2006) Lund
Note
QC 20100712Available from: 2007-12-12 Created: 2007-12-12 Last updated: 2010-07-12Bibliographically approved
3. Sharp spectral inequalitites for the Heisenberg Laplacian
Open this publication in new window or tab >>Sharp spectral inequalitites for the Heisenberg Laplacian
2008 (English)In: Groups and analysis: the legacy of Hermann Weyl / [ed] Tent, Katrin, Cambridge: Cambridge University Press , 2008, 13 sidor- p.Chapter in book (Other academic)
Place, publisher, year, edition, pages
Cambridge: Cambridge University Press, 2008
Series
London Mathematical Society lecture note series, ISSN 0076-0552 ; 354
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7815 (URN)978-0-521-71788-5 (ISBN)
Note
QC 20100712Available from: 2007-12-12 Created: 2007-12-12 Last updated: 2010-07-12Bibliographically approved
4. An inequality between Dirichlet and Neumann eigenvalues of the Heisenberg Laplacian
Open this publication in new window or tab >>An inequality between Dirichlet and Neumann eigenvalues of the Heisenberg Laplacian
2008 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 33, no 12, 2157-2163 p.Article in journal (Refereed) Published
Abstract [en]

Let k and μk be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in Rd, d1. Filonov has proved in a simple way that the inequality μk+1<k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions.

Keyword
Dirichlet-Neumann inequality; Eigenvalue inequality; Heisenberg group; Heisenberg Laplacian; Hypoelliptic; Spectral theory
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7816 (URN)10.1080/03605300802537438 (DOI)000261382300002 ()2-s2.0-57249090385 (Scopus ID)
Note
QC 20100712. Uppdaterad från Submitted till Published 20100712.Available from: 2007-12-12 Created: 2007-12-12 Last updated: 2010-07-12Bibliographically approved

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