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On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
Stockholm University, Faculty of Science, Department of Mathematics.
Purdue University.
2012 (English)In: Computational methods and function theory, ISSN 1617-9447, Vol. 12, no 1, 119-144 p.Article in journal (Refereed) Published
Abstract [en]

We consider the eigenvalue problem with a complex-valued polynomial potentialof arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation.We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions, except for the case d is even and k=d/2 In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.The first results can be derived from H.~Habsch, while the case of a disconnected parameter space is new.

Place, publisher, year, edition, pages
2012. Vol. 12, no 1, 119-144 p.
Keyword [en]
Nevanlinna functions, Schrödinger operator
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-88598OAI: oai:DiVA.org:su-88598DiVA: diva2:612384
Available from: 2013-03-21 Created: 2013-03-21 Last updated: 2013-05-10Bibliographically approved
In thesis
1. Combinatorial Methods in Complex Analysis
Open this publication in new window or tab >>Combinatorial Methods in Complex Analysis
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts.

Part A: Spectral properties of the Schrödinger equation

This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained.

Part B: Graph monomials and sums of squares

In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares.

Part C: Eigenvalue asymptotics of banded Toeplitz matrices

This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above.

Part D: Stretched Schur polynomials

This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2013. 111 p.
Keyword
combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-88808 (URN)978-91-7447-684-2 (ISBN)
Public defence
2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript

Available from: 2013-05-08 Created: 2013-03-30 Last updated: 2013-05-06Bibliographically approved

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Citation style
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