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On Supersingular Perturbations
Stockholm University, Faculty of Science, Department of Mathematics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers and deals with supersingular rank one perturbations of self-adjoint operators and their models in Hilbert or Pontryagin spaces. Here, the term supersingular describes perturbation elements that are outside the underlying space but still obey a certain regularity conditions.

The first two papers study certain Sturm-Liouville differential expressions that can be realised as Schrödinger operators. In Paper I we show that for the potential consisting of the inverse square plus a comparatively well-behaved term we can employ an existing model due to Kurasov to describe these operators in a Hilbert space. In particular, this approach is in good agreement with ODE techniques.

In Paper II we study the inverse fourth power potential.While it is known that the ODE techniques still work, we show that the above model fails and thus that there are limits to the above operator theoretic approach.

In Paper III we concentrate on generalising Kurasov's model. The original formulation assumes that the self-adjoint operator is semi-bounded, whereas we drop this requirement. We give two models with a Hilbert and Pontryagin space structure, respectively, and study the connections between the resulting constructions.

Finally, in Paper IV, we consider the concrete case of the operator of multiplication by the independent variable, a self-adjoint operator whose spectrum covers the real line, and study its perturbations. This illustrates some of the formalism that was developed in the previous paper, and a number of more explicit results are obtained, especially regarding the spectra of the appearing perturbed operators.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2017. , 48 p.
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-147396ISBN: 978-91-7649-982-5 (print)ISBN: 978-91-7649-983-2 (electronic)OAI: oai:DiVA.org:su-147396DiVA: diva2:1144476
Public defence
2017-11-14, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2017-10-20 Created: 2017-09-26 Last updated: 2017-10-04Bibliographically approved
List of papers
1. An Operator Theoretic Interpretation of the Generalized Titchmarsh–Weyl Function for Perturbed Spherical Schrödinger Operators
Open this publication in new window or tab >>An Operator Theoretic Interpretation of the Generalized Titchmarsh–Weyl Function for Perturbed Spherical Schrödinger Operators
2015 (English)In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 9, no 6, 1391-1410 p.Article in journal (Refereed) Published
Abstract [en]

In this paper the operator theoretic interpretation of the generalized Titchmarsh–Weyl function is extended to the perturbed spherical Schrödinger operator.

Keyword
Generalized Titchmarsh–Weyl function, Super singular perturbations, Generalized Nevanlinna function, Singular differential operator
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-114743 (URN)10.1007/s11785-014-0425-8 (DOI)000360086900011 ()
Available from: 2015-03-09 Created: 2015-03-09 Last updated: 2017-09-26Bibliographically approved
2. On the Weyl solution of the 1-dim Schrödinger operator with inverse fourth power potential
Open this publication in new window or tab >>On the Weyl solution of the 1-dim Schrödinger operator with inverse fourth power potential
2016 (English)In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 180, no 2, 295-303 p.Article in journal (Refereed) Published
Abstract [en]

We consider the one dimensional Schrödinger operator with potential 1/x4" role="presentation">1/x4 on the half line. It is known that a generalized Titchmarsh–Weyl function can be associated to it. For other strongly singular potentials in some previous works it was possible to give an operator theoretic interpretation of this fact. However, for the present potential we show that such an interpretation does not exist.

Keyword
Inverse fourth power potential, Schrödinger equation, Strongly singular potential, Titchmarsh–Weyl coefficient, Asymptotic expansion
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-147393 (URN)10.1007/s00605-015-0826-4 (DOI)
Available from: 2017-09-26 Created: 2017-09-26 Last updated: 2017-09-26Bibliographically approved
3. On supersingular perturbations of not necessarily semibounded self-adjoint operators
Open this publication in new window or tab >>On supersingular perturbations of not necessarily semibounded self-adjoint operators
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-147394 (URN)
Available from: 2017-09-26 Created: 2017-09-26 Last updated: 2017-09-26Bibliographically approved
4. Some supersingular perturbations of a multiplication operator
Open this publication in new window or tab >>Some supersingular perturbations of a multiplication operator
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-147395 (URN)
Available from: 2017-09-26 Created: 2017-09-26 Last updated: 2017-09-26Bibliographically approved

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