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Combinatorial Methods in Complex AnalysisPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University , 2013. , 111 p.
##### Keyword [en]

combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-88808ISBN: 978-91-7447-684-2OAI: oai:DiVA.org:su-88808DiVA: diva2:613664
##### Public defence

2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

##### List of papers

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.

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 approved1. On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential$(function(){PrimeFaces.cw("OverlayPanel","overlay612384",{id:"formSmash:j_idt423:0:j_idt427",widgetVar:"overlay612384",target:"formSmash:j_idt423:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential$(function(){PrimeFaces.cw("OverlayPanel","overlay611248",{id:"formSmash:j_idt423:1:j_idt427",widgetVar:"overlay611248",target:"formSmash:j_idt423:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Discriminants, Symmetrized Graph monomials and Sums of Squares$(function(){PrimeFaces.cw("OverlayPanel","overlay581344",{id:"formSmash:j_idt423:2:j_idt427",widgetVar:"overlay581344",target:"formSmash:j_idt423:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Schur polynomials, banded Toeplitz matrices and Widom's formula$(function(){PrimeFaces.cw("OverlayPanel","overlay577130",{id:"formSmash:j_idt423:3:j_idt427",widgetVar:"overlay577130",target:"formSmash:j_idt423:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Around multivariate Schmidt-Spitzer theorem$(function(){PrimeFaces.cw("OverlayPanel","overlay612389",{id:"formSmash:j_idt423:4:j_idt427",widgetVar:"overlay612389",target:"formSmash:j_idt423:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Stretched skew Schur polynomials are recurrent$(function(){PrimeFaces.cw("OverlayPanel","overlay612387",{id:"formSmash:j_idt423:5:j_idt427",widgetVar:"overlay612387",target:"formSmash:j_idt423:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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