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Estimates for Eigenvalues of Schrodinger Operators with Complex-Valued Potentials
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
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
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Text
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. Vol. 106, no 2, p. 197-220
Keyword [en]
Schrodinger operators; polyharmonic operators; complex potential; estimation of eigenvalues
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-125143DOI: 10.1007/s11005-015-0810-xISI: 000368734500003OAI: oai:DiVA.org:liu-125143DiVA: diva2:903221
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2018-02-13
In thesis
1. Resolvent Estimates and Bounds on Eigenvalues for Schrödinger and Dirac Operators
Open this publication in new window or tab >>Resolvent Estimates and Bounds on Eigenvalues for Schrödinger and Dirac Operators
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
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1906
Keyword
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:nbn:se:liu:diva-145173 (URN)978-91-7685-362-7 (ISBN)
Public defence
2018-03-28, BL32, B-huset, Campus Valla, Linköping, 10:15 (English)
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Supervisors
Available from: 2018-02-14 Created: 2018-02-13 Last updated: 2018-02-14Bibliographically approved

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Enblom, Alexandra

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