4,637 research outputs found
Plancherel-Rotach Asymptotics of Second-Order Difference Equations with Linear Coefficients
In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of
polynomials that satisfy a second-order difference equation with linear
coefficients. According to the signs of the parameters, we classify the
difference equations into six cases and derive explicit asymptotic formulas of
the polynomials in the outer and oscillatory regions, respectively. It is
remarkable that the zero distributions of the polynomials may locate on the
imaginary line or even on a sideways Y-shape curve in some cases
Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States
In this paper, we study a family of orthogonal polynomials
arising from nonlinear coherent states in quantum optics. Based on the
three-term recurrence relation only, we obtain a uniform asymptotic expansion
of as the polynomial degree tends to infinity. Our asymptotic
results suggest that the weight function associated with the polynomials has an
unusual singularity, which has never appeared for orthogonal polynomials in the
Askey scheme. Our main technique is the Wang and Wong's difference equation
method. In addition, the limiting zero distribution of the polynomials
is provided
Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems
We study the Plancherel--Rotach asymptotics of four families of orthogonal
polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent
polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials
arise in indeterminate moment problems and three of them are birth and death
process polynomials with cubic or quartic rates. We employ a difference
equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to
a conjecture about large degree behavior of polynomials orthogonal with respect
to solutions of indeterminate moment problems.Comment: 34 pages, typos corrected and references update
Competing electronic orders on Kagome lattices at van Hove filling
The electronic orders in Hubbard models on a Kagome lattice at van Hove
filling are of intense current interest and debate. We study this issue using
the singular-mode functional renormalization group theory. We discover a rich
variety of electronic instabilities under short range interactions. With
increasing on-site repulsion , the system develops successively
ferromagnetism, intra unit-cell antiferromagnetism, and charge bond order. With
nearest-neighbor Coulomb interaction alone (U=0), the system develops
intra-unit-cell charge density wave order for small , s-wave
superconductivity for moderate , and the charge density wave order appears
again for even larger . With both and , we also find spin bond order
and chiral superconductivity in some particular
regimes of the phase diagram. We find that the s-wave superconductivity is a
result of charge density wave fluctuations and the squared logarithmic
divergence in the pairing susceptibility. On the other hand, the d-wave
superconductivity follows from bond order fluctuations that avoid the matrix
element effect. The phase diagram is vastly different from that in honeycomb
lattices because of the geometrical frustration in the Kagome lattice.Comment: 8 pages with 9 color figure
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