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On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 12, no 2, 465-481 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we generalize several results in the article Analytic continuation of eigenvalues of a quartic oscillator of A. Eremenko and A. Gabrielov [4]. We consider a family of eigenvalue problems for a Schrodinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k < (d + 2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k = (d + 2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components.

Place, publisher, year, edition, pages
2012. Vol. 12, no 2, 465-481 p.
Keyword [en]
Nevanlinna functions, Schrodinger operator
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-88370ISI: 000313424300007OAI: oai:DiVA.org:su-88370DiVA: diva2:611248
Note

AuthorCount:1;

Available from: 2013-03-15 Created: 2013-03-13 Last updated: 2017-12-06Bibliographically 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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