Combinatorial Methods in Complex Analysis
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-88808ISBN: 978-91-7447-684-2OAI: oai:DiVA.org:su-88808DiVA: diva2:613664
2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Eremenko, Alexandre, Professor
Shapiro, Boris, Professor
At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript2013-05-082013-03-302013-05-06Bibliographically approved
List of papers