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Schrödinger Operators in Waveguides
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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 [en]
Schrödinger Operators, Hardy Inequalities
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:kth:diva-410ISBN: 91-7178-131-5 (print)OAI: oai:DiVA.org:kth-410DiVA: diva2:10726
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
List of papers
1. Stability of the Magnetic Schrödinger Operator in a Waveguide
Open this publication in new window or tab >>Stability of the Magnetic Schrödinger Operator in a Waveguide
2005 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 30, no 4-6, 539-565 p.Article in journal (Refereed) Published
Abstract [en]

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right.

Keyword
hardy inequality, magnetic field, Schrödinger operator
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-6105 (URN)10.1081/PDE-200050113 (DOI)000230842900004 ()2-s2.0-23444462495 (Scopus ID)
Note
QC 20101007Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2010-10-07Bibliographically approved
2. A Hardy inequality in twisted waveguides
Open this publication in new window or tab >>A Hardy inequality in twisted waveguides
2008 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 188, no 2, 245-264 p.Article in journal (Refereed) Published
Abstract [en]

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.

Keyword
MAGNETIC SCHRÖDINGER OPERATOR, BOUND-STATES, SPECTRUM, EIGENVALUES, CURVATURE, TUBES
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-6106 (URN)10.1007/s00205-007-0106-0 (DOI)000254176700003 ()
Note
QC 20101007. Uppdaterad från accepted till published (20101007).Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2010-10-07Bibliographically approved
3. Lieb-Thirring-type inequalities in a tube with a magnetic field
Open this publication in new window or tab >>Lieb-Thirring-type inequalities in a tube with a magnetic field
(English)Article in journal (Other academic) Submitted
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-6107 (URN)
Note
QS 20120326Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2012-03-26Bibliographically approved
4. Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions
Open this publication in new window or tab >>Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions
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.

National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-6108 (URN)10.1007/s00023-005-0209-9 (DOI)000228868300007 ()2-s2.0-18244384473 (Scopus ID)
Note
QC 20101007Available from: 2005-09-14 Created: 2005-09-14 Last updated: 2010-10-07Bibliographically approved

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