26 research outputs found
Quasi-exactly solvable quartic: elementary integrals and asymptotics
We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where
p, h and P are polynomials in one variable. For the case when h is an odd cubic
polynomial, we found an interesting identity which is used to describe the
spectral locus. We also establish some asymptotic properties of the QES
spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references,
corrected misprint
Quasi-exactly solvable quartic: real algebraic spectral locus
We describe the real quasi-exactly solvable spectral locus of the
PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure
Spherical Quadrilaterals with Three Non-integer Angles
A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that one corner of a quadrilateral is integer (i.e., its angle is a multiple of π) while the angles at its other three corners are not multiples of π. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point
Two-parametric PT-symmetric quartic family
We describe a parametrization of the real spectral locus of the
two-parametric family of PT-symmetric quartic oscillators. For this family, we
find a parameter region where all eigenvalues are real, extending the results
of Dorey, Dunning, Tateo and Shin.Comment: 23 pages, 15 figure
Pole Placement by Static Output Feedback for Generic Linear Systems
We consider linear systems with m inputs, p outputs and McMillan degree n, such that n = mp. If both m and p are even, we show that there is a non-empty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that for each system in U , there exists an open set of pole configurations, symmetric with respect to the real line, which cannot be assigned by any real static output feedback. 1
Degrees of real Wronski maps
We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described