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Around multivariate Schmidt-Spitzer theorem
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 446, 356-368 p.Article 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.

##### Place, publisher, year, edition, pages
2014. Vol. 446, 356-368 p.
##### Keyword [en]
asymptotic root distribution, square and rectangular Toeplitz matrices
Mathematics
Mathematics
##### Identifiers
ISI: 000334146700024OAI: oai:DiVA.org:su-88600DiVA: diva2:612389
Available from: 2013-03-21 Created: 2013-03-21 Last updated: 2015-02-10Bibliographically 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
Mathematics
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)
##### 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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