2,022 research outputs found
Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths
In this paper we consider the model of non-intersecting squared Bessel
processes with parameter , in the confluent case where all particles
start, at time , at the same positive value , remain positive, and
end, at time , at the position . The positions of the paths have a
limiting mean density as which is characterized by a vector
equilibrium problem. We show how to obtain this equilibrium problem from
different considerations involving the recurrence relations for multiple
orthogonal polynomials associated with the modified Bessel functions.
We also extend the situation by rescaling the parameter , letting it
increase proportionally to as increases. In this case we also analyze
the recurrence relation and obtain a vector equilibrium problem for it.Comment: 28 pages, 10 figure
Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions
We consider polynomials P_n orthogonal with respect to the weight J_? on [0,?), where J_? is the Bessel function of order ?. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n?? near the vertical line Rez=??2. We prove this fact for the case 0???1/2 from strong asymptotic formulas that we derive for the polynomials Pn in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for ??1/2
The role of geometry on dispersive forces
The role of geometry on dispersive forces is investigated by calculating the
energy between different spheroidal particles and planar surfaces, both with
arbitrary dielectric properties. The energy is obtained in the non-retarded
limit using a spectral representation formalism and calculating the interaction
between the surface plasmons of the two macroscopic bodies. The energy is a
power-law function of the separation of the bodies, where the exponent value
depends on the geometrical parameters of the system, like the separation
distance between bodies, and the aspect ratio among minor and major axes of the
spheroid.Comment: Presneted at QFEXT05, Barcelona 2005. Submitted to J. Phys.
Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
We establish a relation between several entanglement properties in the
Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins
embedded in a magnetic field. We provide analytical proofs that the single-copy
entanglement and the global geometric entanglement of the ground state close to
and at criticality behave as the entanglement entropy. These results are in
deep contrast to what is found in one- dimensional spin systems where these
three entanglement measures behave differently.Comment: 4 pages, 2 figures, published versio
High-multipolar effects on the Casimir force: the non-retarded limit
We calculate exactly the Casimir force or dispersive force, in the
non-retarded limit, between a spherical nanoparticle and a substrate beyond the
London's or dipolar approximation. We find that the force is a non-monotonic
function of the distance between the sphere and the substrate, such that, it is
enhanced by several orders of magnitude as the sphere approaches the substrate.
Our results do not agree with previous predictions like the Proximity theorem
approach.Comment: 7 pages including 2 figures. Submitted to Europjysics Letter
Universal geometric entanglement close to quantum phase transitions
Under successive Renormalization Group transformations applied to a quantum
state of finite correlation length , there is typically a
loss of entanglement after each iteration. How good it is then to replace
by a product state at every step of the process? In this paper we
give a quantitative answer to this question by providing first analytical and
general proofs that, for translationally invariant quantum systems in one
spatial dimension, the global geometric entanglement per region of size diverges with the correlation length as
close to a quantum critical point with central charge , where is
a cut-off at short distances. Moreover, the situation at criticality is also
discussed and an upper bound on the critical global geometric entanglement is
provided in terms of a logarithmic function of .Comment: 4 pages, 3 figure
Asymptotics of matrix valued orthogonal polynomials on
We analyze the large degree asymptotic behavior of matrix valued orthogonal
polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and
a matrix part. Using the Riemann-Hilbert formulation for MVOPs and the
Deift-Zhou method of steepest descent, we obtain asymptotic expansions for the
MVOPs as the degree tends to infinity, in different regions of the complex
plane (outside the interval of orthogonality, on the interval away from the
endpoints and in neighborhoods of the endpoints), as well as for the matrix
coefficients in the three-term recurrence relation for these MVOPs. The
asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and
Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires
several different factorizations of the matrix part of the weight, in terms of
eigenvalues/eigenvectors and using a matrix Szeg\H{o} function. We illustrate
the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming
from group theory.Comment: 48 pages, 8 figures. Some typos corrected in the last versio
Spectral representation of the Casimir Force Between a Sphere and a Substrate
We calculate the Casimir force in the non-retarded limit between a spherical
nanoparticle and a substrate, and we found that high-multipolar contributions
are very important when the sphere is very close to the substrate. We show that
the highly inhomegenous electromagnetic field induced by the presence of the
substrate, can enhance the Casimir force by orders of magnitude, compared with
the classical dipolar approximation.Comment: 5 page + 4 figures. Submitted to Phys. Rev. Let
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