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Resolvent Estimates and Bounds on Eigenvalues for Schrödinger and Dirac Operators
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-2486-6338
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the spectral theory of Schrödinger and Dirac operators. The main results relate to the problems of estimating perturbed eigenvalues. The thesis is based on four papers.

The first paper focuses on the problem of localization of perturbed eigenvalues for multidimensional Schrödinger operators. Bounds for eigenvalues, lying outside the essential spectrum, are obtained in terms of the Lebesgue's classes. The methods used make it possible to consider the general case of non-self-adjoint operators, and involve the weak Lebesgue's potentials. The results are extended to the case of the polyharmonic operators.

In the second paper, the problem of location of the discrete spectrum is solved for the class of Schrödinger operators considered on the half-line. The general case of complex-valued potentials, imposing various boundary conditions, typically Dirichlet and Neumann conditions, is considered. General mixed boundary conditions are also treated.

The third paper is devoted to Dirac operators. The case of spherically symmetric potentials is considered. Estimates for the eigenvalues are derived from the asymptotic behaviour of the resolvent of the free Dirac operator. For the massless Dirac operators, whose essential spectrum is the whole real line, optimal bounds for the imaginary part of the eigenvalues are established.

In the fourth paper, new Hardy-Carleman type inequalities for Dirac operators are proven. Concrete Carleman type inequalities, useful in applications, Agmon and also Treve type inequalities are derived from the general results by involving special weight functions. The results are extended to the case of the Dirac operator describing the relativistic particle in a potential magnetic field.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. , p. 39
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1906
Keywords [en]
Spectral theory, Schrödinger operators, polyharmonic operators, Dirac operators, non-self-adjoint perturbations, complex potential, estimation of eigenvalues, Carleman inequalities, Hardy inequalities
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-145173DOI: 10.3384/diss.diva-145173ISBN: 9789176853627 (print)OAI: oai:DiVA.org:liu-145173DiVA, id: diva2:1182325
Public defence
2018-03-28, BL32, B-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2018-02-14 Created: 2018-02-13 Last updated: 2018-05-17Bibliographically approved
List of papers
1. Estimates for Eigenvalues of Schrodinger Operators with Complex-Valued Potentials
Open this publication in new window or tab >>Estimates for Eigenvalues of Schrodinger Operators with Complex-Valued Potentials
2016 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 106, no 2, p. 197-220Article in journal (Refereed) Published
Abstract [en]

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrodinger operators are obtained in terms of L (p) -norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745-750, 2011), Safronov (Proc Am Math Soc 138(6):2107-2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29-54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrodinger operators with slowly decaying potentials and belonging to weak Lebesgues classes are also considered.

Place, publisher, year, edition, pages
SPRINGER, 2016
Keywords
Schrodinger operators; polyharmonic operators; complex potential; estimation of eigenvalues
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-125143 (URN)10.1007/s11005-015-0810-x (DOI)000368734500003 ()
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2018-02-13
2. ESTIMATES OF EIGENVALUES OF SCHRODINGER OPERATORS ON THE HALF-LINE WITH COMPLEX-VALUED POTENTIALS
Open this publication in new window or tab >>ESTIMATES OF EIGENVALUES OF SCHRODINGER OPERATORS ON THE HALF-LINE WITH COMPLEX-VALUED POTENTIALS
2017 (English)In: Operators and Matrices, ISSN 1846-3886, E-ISSN 1848-9974, Vol. 11, no 2, p. 369-380Article in journal (Refereed) Published
Abstract [en]

Estimates for eigenvalues of Schrodinger operators on the half-line with complex-valued potentials are established. Schrodinger operators with potentials belonging to weak Lebesques classes are also considered. The results cover those known previously due to R. L. Frank, A. Laptev and R. Seiringer

Place, publisher, year, edition, pages
ELEMENT, 2017
Keywords
Schrodinger operators; complex potentials; estimation of eigenvalues
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-142843 (URN)10.7153/oam-11-25 (DOI)000413116800006 ()
Available from: 2017-11-06 Created: 2017-11-06 Last updated: 2018-02-13
3. Resolvent estimates and bounds on eigenvalues for Dirac operators on the half-line
Open this publication in new window or tab >>Resolvent estimates and bounds on eigenvalues for Dirac operators on the half-line
2018 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 16, article id 165203Article in journal (Refereed) Epub ahead of print
Abstract [en]

Estimates for the eigenvalues of non-self-adjoint 1D Dirac operators considered on the half-line are obtained in terms of the L p -norms of the potentials. The proofs are based on the resolvent estimates established for the free Dirac operator.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2018
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147850 (URN)10.1088/1751-8121/aab487 (DOI)
Available from: 2018-05-16 Created: 2018-05-16 Last updated: 2018-05-17Bibliographically approved
4. Hardy-Carleman type inequalities for Dirac operators
Open this publication in new window or tab >>Hardy-Carleman type inequalities for Dirac operators
2015 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 56, no 10, p. 103503-Article in journal (Refereed) Published
Abstract [en]

General Hardy-Carleman type inequalities for Dirac operators are proved. New inequalities are derived involving particular traditionally used weight functions. In particular, a version of the Agmon inequality and Treve type inequalities is established. The case of a Dirac particle in a (potential) magnetic field is also considered. The methods used are direct and based on quadratic form techniques. (C) 2015 AIP Publishing LLC.

Place, publisher, year, edition, pages
AMER INST PHYSICS, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123075 (URN)10.1063/1.4933241 (DOI)000364237000030 ()
Available from: 2015-12-04 Created: 2015-12-03 Last updated: 2018-02-13

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