Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior
Lund University.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Dipartimento di Matematica, Politecnico di Torino.
2010 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 248, no 4, 850-865 p.Article in journal (Refereed) Published
Abstract [en]

Consider a regular d-dimensional metric tree Γ with root o. Define the Schrödinger operator - Δ - V, where V is a non-negative, symmetric potential, on Γ, with Neumann boundary conditions at o. Provided that V decays like | x |- γ at infinity, where 1 < γ ≤ d ≤ 2, γ ≠ 2, we will determine the weak coupling behavior of the bottom of the spectrum of - Δ - V. In other words, we will describe the asymptotic behavior of inf σ (- Δ - α V) as α → 0 +.

Place, publisher, year, edition, pages
2010. Vol. 248, no 4, 850-865 p.
Keyword [en]
Fourier-Bessel transformation; Metric trees; Schrödinger operators; Weak coupling
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-9979DOI: 10.1016/j.jde.2009.11.011ISI: 000274197200009Scopus ID: 2-s2.0-73549101079OAI: oai:DiVA.org:kth-9979DiVA: diva2:173795
Note

QC 20100712. Updated from manuscript to article in journal 9 December 2009. QC 20111116

Available from: 2009-02-17 Created: 2009-02-17 Last updated: 2012-08-28Bibliographically approved
In thesis
1. Properties of the Discrete and Continuous Spectrum of Differential Operators
Open this publication in new window or tab >>Properties of the Discrete and Continuous Spectrum of Differential Operators
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three scientific papers devoted to the study of different spectral theoretical aspects of differential operators in Hilbert spaces.The first paper concerns the magnetic Schrödinger operator (i∇ + A)2 in L2(ℝn). It is proved that given certain conditions on the decay of A, the set [0,∞) is an essential support of the absolutely continuous part of the spectral measure corresponding to the operator.The second paper considers a regular d-dimensional metric tree Γ and defines Schrödinger operators - Δ - V on it.  Here, V is a symmetric, non-negative potential on Γ. It is assumed that V decays like lxl at infinity, where 1 < γ ≤ d ≤2, γ ≠ 2. A weak coupling constant α is introduced in front of V, and the asymptotics of the bottom of the spectrum as α → 0+ is described.The third, and last, paper revolves around fourth-order differential operators in the space L2(ℝn), where n = 1 or n = 3.  In particular, the operator (-Δ)2 - C|x|-4 - V(x) is studied, where C is the sharp constant in the Hardy-Rellich inequality. A Lieb-Thirring inequality for this operator is proved, and as a consequence a Sobolev-type inequality is obtained.

 

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vii, 30 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 09:02
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9981 (URN)978-91-7415-227-2 (ISBN)
Public defence
2009-02-27, F3, KTH, Lindstedtsvägen 26, Stockholm, 14:00 (English)
Opponent
Supervisors
Note
QC 20100712Available from: 2009-02-18 Created: 2009-02-17 Last updated: 2010-07-12Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full textScopushttp://dx.doi.org/10.1016/j.jde.2009.11.011

Search in DiVA

By author/editor
Ekholm, TomasEnblom, Andreas
By organisation
Mathematics (Div.)
In the same journal
Journal of Differential Equations
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 66 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf