5,593 research outputs found
Elementary proof of the B. and M. Shapiro conjecture for rational functions
We give a new elementary proof of the following theorem: if all critical
points of a rational function g belong to the real line then there exists a
fractional linear transformation L such that L(g) is a real rational function.
Then we interpret the result in terms of Fuchsian differential equations whose
general solution is a polynomial and in terms of electrostatics.Comment: 21 page
Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families
In the first part of the paper, we discuss eigenvalue problems of the form
-w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on
two rays in the complex plane. We give sufficient conditions for continuity of
the spectrum when the leading coefficient of P tends to 0. In the second part,
we apply these results to the study of topology and geometry of the real
spectral loci of PT-symmetric families with P of degree 3 and 4, and prove
several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved,
generalized, and the proof is simplified. 37 pages, 16 figure
On metrics of curvature 1 with four conic singularities on tori and on the sphere
We discuss conformal metrics of curvature 1 on tori and on the sphere, with
four conic singularities whose angles are multiples of pi/2. Besides some
general results we study in detail the family of such symmetric metrics on the
sphere, with angles (pi/2,3pi/2,pi/2,3pi/2).Comment: 25 pages, 5 figure
Metrics with four conic singularities and spherical quadrilaterals
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 two angles at the corners are multiples of pi. The
problem is equivalent to classification of Heun's equations with real
parameters and unitary monodromy.Comment: 68 pges, 25 figure
Zeros of eigenfunctions of some anharmonic oscillators
We study eigenfunctions of Schrodinger operators -y"+Py on the real line with
zero boundary conditions, whose potentials P are real even polynomials with
positive leading coefficients. For quartic potentials we prove that all zeros
of all eigenfunctions belong to the union of the real and imaginary axes.
Similar result holds for sextic potentials and their eigenfunctions with
finitely many complex zeros. As a byproduct we obtain a complete classification
of such eigenfunctions of sextic potentials.Comment: 22 pages 8 figure
High energy eigenfunctions of one-dimensional Schrodinger operators with polynomial potentials
For a class of one-dimensional Schrodinger operators with polynomial
potentials that includes Hermitian and PT-symmetric operators, we show that the
zeros of scaled eigenfunctions have a limit disctibution in the complex plane
as the eigenvalues tend to infinity.
This limit distribution depends only on the degree of potential and on the
boundary conditions.Comment: 22 pages, 9 figure
Spherical quadrilaterals with three non-integer angles
We classify spherical quadrilaterals up to isometry in the case when one
inner angle is a multiple of pi while the other three are not. This is
equivalent to classification of Heun's equations with real parameters and one
apparent singularity such that the monodromy consists of unitary
transformations.Comment: 38 pages, 16 figures. arXiv admin note: text overlap with
arXiv:1409.152
- …