Properties of the Discrete and Continuous Spectrum of Differential Operators
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
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-Γ 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.
Trita-MAT. MA, ISSN 1401-2278 ; 09:02
IdentifiersURN: urn:nbn:se:kth:diva-9981ISBN: 978-91-7415-227-2OAI: oai:DiVA.org:kth-9981DiVA: diva2:173814
2009-02-27, F3, KTH, Lindstedtsvägen 26, Stockholm, 14:00 (English)
Ilyin, Alexei, Professor
Laptev, Ari, Professor
QC 201007122009-02-182009-02-172010-07-12Bibliographically approved
List of papers