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On Lieb-Thirring inequalities for Schrödinger operators with virtual level
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2006 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, 725-740 p.Article in journal (Refereed) Published
Abstract [en]

We consider the operator H = - Delta- V in L-2(R-d), d >= 3. For the moments of its negative eigenvalues we prove the estimate

tr H--(gamma) <= C-gamma,C-d integral(Rd) (V(x) - (d-2)(2)/4\x\(2))(gamma+d/2) dx, gamma > 0.

Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.

Place, publisher, year, edition, pages
2006. Vol. 264, no 3, 725-740 p.
Keyword [en]
BOUND-STATES; NUMBER
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7026DOI: 10.1007/s00220-006-1521-zISI: 000237193800008Scopus ID: 2-s2.0-33646531642OAI: oai:DiVA.org:kth-7026DiVA: diva2:11903
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2010-07-08Bibliographically approved
In thesis
1. Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators
Open this publication in new window or tab >>Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators.

In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigen¬values remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator.

In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter. In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclu¬sion of magnetic fields.

As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge.

In Paper V we derive inequalities for moments of the real part and the modulus of the eigen¬values of Schrödinger operators with complex-valued potentials.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. vii, 28 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 07:03
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4344 (URN)978-91-7178-626-5 (ISBN)
Public defence
2007-05-09, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 09:00
Opponent
Supervisors
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2010-07-08Bibliographically approved

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