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Properties of the Discrete and Continuous Spectrum of Differential Operators
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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: urn:nbn:se:kth:diva-9981ISBN: 978-91-7415-227-2 (print)OAI: oai:DiVA.org:kth-9981DiVA: diva2:173814
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
List of papers
1. On the absolutely continuous spectrum of magnetic Schrödinger operators
Open this publication in new window or tab >>On the absolutely continuous spectrum of magnetic Schrödinger operators
2009 (English)In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 156, no 4, 699-723 p.Article in journal (Refereed) Published
Abstract [en]

We prove that [0, ∞) is an essential support for the absolutely continuous part of the spectral measure associated with the magnetic Schrödinger operator (i▽+ A)2 in L2(ℝ ν), given certain conditions on the decay of A. Bibliography: 8 titles. Illustrations: 1 figure.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9980 (URN)10.1007/s10958-009-9282-9 (DOI)2-s2.0-59849096244 (Scopus ID)
Note
QC 20100712Available from: 2009-02-17 Created: 2009-02-17 Last updated: 2017-12-13Bibliographically approved
2. Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior
Open this publication in new window or tab >>Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior
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 +.

Keyword
Fourier-Bessel transformation; Metric trees; Schrödinger operators; Weak coupling
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9979 (URN)10.1016/j.jde.2009.11.011 (DOI)000274197200009 ()2-s2.0-73549101079 (Scopus ID)
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: 2017-12-13Bibliographically approved
3. Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions
Open this publication in new window or tab >>Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions
(English)Manuscript (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9978 (URN)
Note
QC 20100712Available from: 2009-02-17 Created: 2009-02-17 Last updated: 2012-01-27Bibliographically approved

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Citation style
  • apa
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