arXiv subjects

Per Alexandersson

Publications and source records attributed to Per Alexandersson.

58 records · Page 4Linked to original sources

arXiv2012

Schur polynomials, banded Toeplitz matrices and Widom's formula

We prove that for arbitrary partitions $\mathbfλ \subseteq \mathbfκ,$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbfκ + k\cdot \mathbf{1}^c)/(\mathbfλ + k\cdot \mathbf{1}^r)}(x_1,...,x_n)$ 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.

math.CO↗

arXiv2012

Discriminants, symmetrized graph monomials, and sums of squares certificates

Here we present certificates for 5 classes of 6-edged multigraphs whose symmetrized graph monomials may be represented as sum of squares, but not as linear combinations of partition square graphs. This is a complement to the results presented in Discriminants, symmetrized graph monomials, and sums of squares.

math.AG↗

arXiv2011

On eigenvalues of the Schrödinger operator with an even complex-valued polynomial potential

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schrödinger 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.

math-ph↗

arXiv2011

Discriminants, symmetrized graph monomials, and sums of squares

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.

math.AG↗