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Spectral Inequalities and Their Applications in Quantum Mechanics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work presented in this thesis revolves around spectral inequalities and their applications in quantum mechanics.

In Paper A, the ground state energy of an atom confined to two dimensions is analyzed in the limit when the charge of the nucleus Z becomes very large. The main result is a two-term asymptotic expansion of the ground state energy in terms of Z.

Paper B deals with Hardy inequalities for the kinetic energy of a particle in the presence of an external magnetic field. If the magnetic field has a non-trivial radial component, we show that Hardy’s classical lower bound can be improved by an extra term depending on the magnetic field.

In Paper C we study interacting Bose gases and prove Lieb-Thirring type estimates for several types of interaction potentials, such as the hard-sphere interaction in three dimensions, the hard-disk interaction in two dimensions as well as homogeneous potentials. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , viii, 34 p.
Series
TRITA-MAT-A, 2014:08
Keyword [en]
Spectral inequalities, quantum mechanics, Hardy inequalities, Sobolev inequalities, Lieb-Thirring inequalities, stability of matter, Bose gases, Bose-Einstein condensation
National Category
Mathematical Analysis
Research subject
Mathematics; Physics
Identifiers
URN: urn:nbn:se:kth:diva-145210ISBN: 978-91-7595-168-3 (print)OAI: oai:DiVA.org:kth-145210DiVA: diva2:717323
Public defence
2014-06-05, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2008-5048Swedish Research Council, 2012-3864
Note

QC 20140520

Available from: 2014-05-20 Created: 2014-05-14 Last updated: 2014-05-20Bibliographically approved
List of papers
1. Asymptotics for Two-Dimensional Atoms
Open this publication in new window or tab >>Asymptotics for Two-Dimensional Atoms
2012 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 13, no 2, 333-362 p.Article in journal (Refereed) Published
Abstract [en]

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z > 0 and N quantum electrons of charge -1 is E(N, Z) = - 1/2 Z(2) lnZ + ( E-TF (lambda) + 1/2 c(H))Z(2) + o(Z(2)) when Z -> a and N/Z -> lambda, where E (TF)(lambda) is given by a Thomas-Fermi type variational problem and c (H) a parts per thousand -2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z -> a, which is contrary to the expected behavior of three-dimensional atoms.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-91245 (URN)10.1007/s00023-011-0123-2 (DOI)000300279900004 ()2-s2.0-84856744468 (Scopus ID)
Note
QC 20120312Available from: 2012-03-12 Created: 2012-03-12 Last updated: 2017-12-07Bibliographically approved
2. A magnetic contribution to the Hardy inequality
Open this publication in new window or tab >>A magnetic contribution to the Hardy inequality
2014 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 55, no 2, 022101- p.Article in journal (Refereed) Published
Abstract [en]

We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.

Keyword
Schrödinger Operators, Dirichlet Forms
National Category
Mathematical Analysis
Research subject
Mathematics; Physics
Identifiers
urn:nbn:se:kth:diva-144129 (URN)10.1063/1.4863900 (DOI)000332486500008 ()2-s2.0-84902244627 (Scopus ID)
Funder
Swedish Research Council, 2009-6073Swedish Research Council, 2008-5048
Note

QC 20140624

Available from: 2014-04-14 Created: 2014-04-10 Last updated: 2017-12-05Bibliographically approved
3. Lieb-Thirring Bounds for Interacting Bose Gases
Open this publication in new window or tab >>Lieb-Thirring Bounds for Interacting Bose Gases
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 335, no 2, 1019-1056 p.Article in journal (Refereed) Published
Abstract [en]

We study interacting Bose gases and prove lower bounds for the kinetic plus interaction energy of a many-body wave function in terms of its particle density. These general estimates are then applied to various types of interactions, including hard sphere (in 3D) and hard disk (in 2D) as well as a general class of homogeneous potentials.

Keyword
Interacting Bose gas, quantum many-body problem, energy inequalities, Lieb-Thirring inequalities, local exclusion principle, local uncertainty principle, hard-sphere interaction, hard-disk interaction, homogeneous potentials, scattering length
National Category
Mathematical Analysis Condensed Matter Physics
Research subject
Mathematics; Physics
Identifiers
urn:nbn:se:kth:diva-145067 (URN)10.1007/s00220-014-2278-4 (DOI)000350367700015 ()2-s2.0-84925493862 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2010.0063Swedish Research Council, 2013-4734 2012-3864EU, European Research Council, 321029
Note

QC 20150408. Updated from manuscript to article in journal.

Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2017-12-05Bibliographically approved

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