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Combinatorial Methods in Complex Analysis
Stockholm University, Faculty of Science, Department of Mathematics.
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. , p. 111
##### Keyword [en]
combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
Mathematics
Mathematics
##### Identifiers
ISBN: 978-91-7447-684-2 (print)OAI: oai:DiVA.org:su-88808DiVA: diva2:613664
##### Public defence
2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### 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At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript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
##### List of papers
1. On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
Open this publication in new window or tab >>On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential
2012 (English)In: Computational methods and function theory, ISSN 1617-9447, Vol. 12, no 1, p. 119-144Article 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.

##### Keyword
Nevanlinna functions, Schrödinger operator
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:su:diva-88598 (URN)
Available from: 2013-03-21 Created: 2013-03-21 Last updated: 2013-05-10Bibliographically approved
2. On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential
Open this publication in new window or tab >>On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential
2012 (English)In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 12, no 2, p. 465-481Article 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.

##### Keyword
Nevanlinna functions, Schrodinger operator
Mathematics
##### Identifiers
urn:nbn:se:su:diva-88370 (URN)000313424300007 ()
##### Note

AuthorCount:1;

Available from: 2013-03-15 Created: 2013-03-13 Last updated: 2017-12-06Bibliographically approved
3. Discriminants, Symmetrized Graph monomials and Sums of Squares
Open this publication in new window or tab >>Discriminants, Symmetrized Graph monomials and Sums of Squares
2012 (English)In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 21, no 4, p. 353-361Article in journal (Refereed) Published
##### Abstract [en]

In 1878, motivated by the requirements of the invariant the-ory of binary forms, J. J. Sylvester constructed, for every graphwith possible multiple edges but without loops, its symmetrizedgraph monomial, which is a polynomial in the vertex labels ofthe original graph. We pose the question for which graphs thispolynomial is nonnegative or a sum of squares. This problem ismotivated by a recent conjecture of F. Sottile and E. Mukhin onthe discriminant of the derivative of a univariate polynomial andby an interesting example of P. and A. Lax of a graph with fouredges whose symmetrized graph monomial is nonnegative butnot a sum of squares. We present detailed information about sym-metrized graph monomials for graphs with four and six edges,obtained by computer calculations.

##### Keyword
polynomial sums of squares, translation-invariant polynomials, graph monomials, sums of squares, discriminants, symmetric polynomials
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:su:diva-84745 (URN)10.1080/10586458.2012.669608 (DOI)000313614400003 ()
Available from: 2012-12-31 Created: 2012-12-31 Last updated: 2017-12-06Bibliographically approved
4. Schur polynomials, banded Toeplitz matrices and Widom's formula
Open this publication in new window or tab >>Schur polynomials, banded Toeplitz matrices and Widom's formula
2012 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, no 4, p. P22-Article in journal (Refereed) Published
##### Abstract [en]

We prove that for arbitrary partitions lambda subset of kappa, and integers 0 <= c < r <= n, the sequence of Schur polynomials S(kappa+k.1c)/(lambda+k.1r)(x(1), ... , x(n)) for k sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

##### Keyword
Banded Toeplitz matrices, Schur polynomials, Widom's determinant formula, sequence insertion, Young tableaux, recurrence
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:su:diva-83811 (URN)000310833500004 ()
##### Note

AuthorCount:1;

Available from: 2012-12-19 Created: 2012-12-14 Last updated: 2017-12-06Bibliographically approved
5. Around multivariate Schmidt-Spitzer theorem
Open this publication in new window or tab >>Around multivariate Schmidt-Spitzer theorem
2014 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 446, p. 356-368Article in journal (Refereed) Published
##### Abstract [en]

Given an arbitrary complex-valued infinite matrix $\infmatA=(a_{ij}),$$i=1,\dotsc,\infty;$ $j=1,\dotsc,\infty$  and a positive integer $n$ we introduce anaturally associated  polynomial basis $\polybasis_\infmatA$ of$\C[x_0,\dotsc,x_n]$.We discuss some properties of the locus of  common zeros of all polynomials in $\polybasis_A$ having  a given degree $m$; the latter locus can beinterpreted as the spectrum of the $m\times (m+n)$-submatrix of $\infmatA$ formed by its  $m$ first rows and$(m+n)$ first columns. We initiate the study of the asymptotics of these spectra when $m\to \infty$ inthe case when $\infmatA$ is a banded Toeplitz matrix.In particular, we present and partially prove a conjectural multivariate analogof the well-known Schmidt-Spitzer theorem which describes  the spectral asymptotics for the sequence of principal minors of an arbitrarybanded Toeplitz matrix.Finally, we discuss relations between polynomial bases $\polybasis_\infmatA$ andmultivariate  orthogonal polynomials.

##### Keyword
asymptotic root distribution, square and rectangular Toeplitz matrices
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:su:diva-88600 (URN)10.1016/j.laa.2014.01.005 (DOI)000334146700024 ()
Available from: 2013-03-21 Created: 2013-03-21 Last updated: 2017-11-22Bibliographically approved
6. Stretched skew Schur polynomials are recurrent
Open this publication in new window or tab >>Stretched skew Schur polynomials are recurrent
2014 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 122, p. 1-8Article in journal (Refereed) Published
##### Abstract [en]

We show that sequences of skew Schur polynomials obtained from stretched semi-standard Young tableauxsatisfy a linear recurrence, which we give explicitly.Using this, we apply this to finding certain asymptotic behavior of these Schur polynomials and present conjectures on minimal recurrences for stretched Schur polynomials.

##### Keyword
Schur polynomials, tableau concatenation, Young tableaux, recurrence, asymptotics
Mathematics
Mathematics
##### Identifiers
urn:nbn:se:su:diva-88599 (URN)10.1016/j.jcta.2013.09.009 (DOI)000327416300001 ()
Available from: 2013-03-21 Created: 2013-03-21 Last updated: 2017-12-06Bibliographically approved

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