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
Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Faculty of Mathematics and Physics, Stuttgart University.
2005 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 6, no 2, 327-342 p.Article in journal (Refereed) Published
Abstract [en]

We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann(1). We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.

Place, publisher, year, edition, pages
2005. Vol. 6, no 2, 327-342 p.
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:kth:diva-6108DOI: 10.1007/s00023-005-0209-9ISI: 000228868300007Scopus ID: 2-s2.0-18244384473OAI: oai:DiVA.org:kth-6108DiVA: diva2:10725
Note
QC 20101007Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Schrödinger Operators in Waveguides
Open this publication in new window or tab >>Schrödinger Operators in Waveguides
2005 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis, which consists of four papers, we study the discrete spectrum of Schrödinger operators in waveguides. In these domains the quadratic form of the Dirichlet Laplacian operator does not satisfy any Hardy inequality. If we include an attractive electric potential in the model or curve the domain, then bound states will always occur with energy below the bottom of the essential spectrum. We prove that a magnetic field stabilises the threshold of the essential spectrum against small perturbations. We deduce this fact from a magnetic Hardy inequality, which has many interesting applications in itself.

In Paper I we prove the magnetic Hardy inequality in a two-dimensional waveguide. As an application, we establish that when a magnetic field is present, a small local deformation or a small local bending of the waveguide will not create bound states below the essential spectrum.

In Paper II we study the Dirichlet Laplacian operator in a three-dimensional waveguide, whose cross-section is not rotationally invariant. We prove that if the waveguide is locally twisted, then the lower edge of the spectrum becomes stable. We deduce this from a Hardy inequality.

In Paper III we consider the magnetic Schrödinger operator in a three-dimensional waveguide with circular cross-section. If we include an attractive potential, eigenvalues may occur below the bottom of the essential spectrum. We prove a magnetic Lieb-Thirring inequality for these eigenvalues. In the same paper we give a lower bound on the ground state of the magnetic Schrödinger operator in a disc. This lower bound is used to prove a Hardy inequality for the magnetic Schrödinger operator in the original waveguide setting.

In Paper IV we again study the two-dimensional waveguide. It is known that if the boundary condition is changed locally from Dirichlet to magnetic Neumann, then without a magnetic field bound states will occur with energies below the essential spectrum. We however prove that in the presence of a magnetic field, there is a critical minimal length of the magnetic Neumann boundary condition above which the system exhibits bound states below the threshold of the essential spectrum. We also give explicit bounds on the critical length from above and below.

Place, publisher, year, edition, pages
Stockholm: KTH, 2005. ix, 16 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 05:10
Keyword
Schrödinger Operators, Hardy Inequalities
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-410 (URN)91-7178-131-5 (ISBN)
Public defence
2005-09-23, D3, KTH, Lindstedtvägen 5, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20101007Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2010-10-07Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Ekholm, Tomas
By organisation
Mathematics (Dept.)
In the same journal
Annales de l'Institute Henri Poincare. Physique theorique
Other Physics Topics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 80 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