Densities in Fabry's theorem
Fabry's theorem on the singularities of power series is improved: the maximum density in the assumptions of this theorem is replaced by an interior density of Beurling--Malliavin type.
math.CV↗
arXiv subjects
Alexandre Eremenko
Publications and source records attributed to Alexandre Eremenko.
Tangencies between holomorphic maps and holomorphic laminations
We prove that the set of leaves of a holomorphic lamination of codimension one that are non-transversal to a germ of a holomorphic map is discrete.
math.CV↗
Brody curves omitting hyperplanes
A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex line to the complex projective space of dimension n, such that the family of its translations is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n=1.
math.CV↗
Rational functions with real multipliers
Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.
math.DS↗
Meromorphic functions with linearly distributed values and Julia sets of rational functions
If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle.
math.CV↗
An extremal problem for a class of entire functions of exponential type
We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.
math.CV↗
A version of Fabry's theorem for power series with regularly varying coefficients
For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.
math.CV↗
Analytic continuation of eigenvalues of a quartic oscillator
We consider the Schrodinger operator on the real line with even quartic potential and study analytic continuation of eigenvalues, as functions of the coefficient of the potential. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the plane.
math-ph↗
Normal holomorphic curves from parabolic regions to projective spaces
A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems.
math.CV↗
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.
math-ph↗
On nodal sets and nodal domains on S^2 and R^2
We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S^2. We also construct a solution of the equation Delta u=u in R^2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
math.SP↗
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.
math-ph↗
Direct singularities and completely invariant domains of entire functions
Let f be a transcendental entire function that omits a complex value a. We show that for every simply connected region D that does not contain a the full preimage of D is disconnected. We conjecture that the same holds if one only assumes that a is omitted locally. We were able to prove this conjecture under the additional assumption that f is of finite order. We include some auxilliary results on the singularities of the inverses of entire functions which are of independent interest.
math.CV↗
Proof of a conjecture of Polya on the zeros of successive derivatives of real entire functions
We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.
math.CV↗
A Markov-type inequality for arbitrary plane continua
A version of Markov's estimate for the derivative of a polynomial is proved with the interval [-1,1] replaced by an arbitrary continuum in the complex plane.
math.CV↗
Uniform approximation of sgn(x) by polynomials and entire functions
We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the best approximation as the degrees tend to infinity. Our approach is based on a new representation of the extremal polynomials in terms of certain conformal maps. We also study the best polynomial approximation of sgn(x) on [-1,1] with respect to the Hausdorff distance between the completed graphs.
math.CA↗
Meromorphic functions with two completely invariant domains
We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia set need not be a quasicircle. We also show that all critical and asymptotic values are contained in the two completely invariant components. This need not be the case for functions with infinitely many critical and asymptotic values.
math.CV↗
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.
math.AG↗