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Asymptotics for Two-Dimensional Atoms
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2012. Vol. 13, no 2, 333-362 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-91245DOI: 10.1007/s00023-011-0123-2ISI: 000300279900004Scopus ID: 2-s2.0-84856744468OAI: oai:DiVA.org:kth-91245DiVA: diva2:509297
Note
QC 20120312Available from: 2012-03-12 Created: 2012-03-12 Last updated: 2017-12-07Bibliographically approved
In thesis
1. Spectral Inequalities and Their Applications in Quantum Mechanics
Open this publication in new window or tab >>Spectral Inequalities and Their Applications in Quantum Mechanics
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
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:nbn:se:kth:diva-145210 (URN)978-91-7595-168-3 (ISBN)
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

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