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. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript</p>
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:su-88808 |
| Date | January 2013 |
| Creators | Alexandersson, Per |
| Publisher | Stockholms universitet, Matematiska institutionen, Stockholm : Department of Mathematics, Stockholm University |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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