375 research outputs found
Orthogonal polynomials for the weakly equilibrium Cantor sets
Let be the weakly equilibrium Cantor type set introduced in [10].
It is proven that the monic orthogonal polynomials with respect to
the equilibrium measure of coincide with the Chebyshev polynomials
of the set. Procedures are suggested to find of all degrees and the
corresponding Jacobi parameters. It is shown that the sequence of the Widom
factors is bounded below
Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings
Let be a probability measure with an infinite compact support on
. Let us further assume that is a sequence of
orthogonal polynomials for where is a sequence of
nonlinear polynomials and for all
. We prove that if there is an such that
is a root of for each then the distance between any two
zeros of an orthogonal polynomial for of a given degree greater than
has a lower bound in terms of the distance between the set of critical points
and the set of zeros of some . Using this, we find sharp bounds from below
and above for the infimum of distances between the consecutive zeros of
orthogonal polynomials for singular continuous measures.Comment: Contains less typo
Orthogonal polynomials on generalized Julia sets
We extend results by Barnsley et al. about orthogonal polynomials on Julia
sets to the case of generalized Julia sets. The equilibrium measure is
considered. In addition, we discuss optimal smoothness of Green functions and
Parreau-Widom criterion for a special family of real generalized Julia sets.Comment: We changed the second part of the article a little bit and gave
sharper results in this versio
Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties
Hamiltonian tridiagonal matrices characterized by multi-fractal spectral
measures in the family of Iterated Function Systems can be constructed by a
recursive technique here described. We prove that these Hamiltonians are
almost-periodic. They are suited to describe quantum lattice systems with
nearest neighbours coupling, as well as chains of linear classical oscillators,
and electrical transmission lines.
We investigate numerically and theoretically the time dynamics of the systems
so constructed. We derive a relation linking the long-time, power-law behaviour
of the moments of the position operator, expressed by a scaling function
of the moment order , and spectral multi-fractal dimensions,
D_q, via . We show cases in which this relation
is exact, and cases where it is only approximate, unveiling the reasons for the
discrepancies.Comment: 13 pages, Latex, 6 postscript figures. Accepted for publication in
Physica
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