1,026 research outputs found
Meixner polynomials of the second kind and quantum algebras representing su(1,1)
We show how Viennot's combinatorial theory of orthogonal polynomials may be
used to generalize some recent results of Sukumar and Hodges on the matrix
entries in powers of certain operators in a representation of su(1,1). Our
results link these calculations to finding the moments and inverse polynomial
coefficients of certain Laguerre polynomials and Meixner polynomials of the
second kind. As an immediate consequence of results by Koelink, Groenevelt and
Van Der Jeugt, for the related operators, substitutions into essentially the
same Laguerre polynomials and Meixner polynomials of the second kind may be
used to express their eigenvectors. Our combinatorial approach explains and
generalizes this "coincidence".Comment: several correction
Nucleus-Electron Model for States Changing from a Liquid Metal to a Plasma and the Saha Equation
We extend the quantal hypernetted-chain (QHNC) method, which has been proved
to yield accurate results for liquid metals, to treat a partially ionized
plasma. In a plasma, the electrons change from a quantum to a classical fluid
gradually with increasing temperature; the QHNC method applied to the electron
gas is in fact able to provide the electron-electron correlation at arbitrary
temperature. As an illustrating example of this approach, we investigate how
liquid rubidium becomes a plasma by increasing the temperature from 0 to 30 eV
at a fixed normal ion-density . The electron-ion
radial distribution function (RDF) in liquid Rb has distinct inner-core and
outer-core parts. Even at a temperature of 1 eV, this clear distinction remains
as a characteristic of a liquid metal. At a temperature of 3 eV, this
distinction disappears, and rubidium becomes a plasma with the ionization 1.21.
The temperature variations of bound levels in each ion and the average
ionization are calculated in Rb plasmas at the same time. Using the
density-functional theory, we also derive the Saha equation applicable even to
a high-density plasma at low temperatures. The QHNC method provides a procedure
to solve this Saha equation with ease by using a recursive formula; the charge
population of differently ionized species are obtained in Rb plasmas at several
temperatures. In this way, it is shown that, with the atomic number as the only
input, the QHNC method produces the average ionization, the electron-ion and
ion-ion RDF's, and the charge population which are consistent with the atomic
structure of each ion for a partially ionized plasma.Comment: 28 pages(TeX) and 11 figures (PS
One-parameter extension of the Doi-Peliti formalism and relation with orthogonal polynomials
An extension of the Doi-Peliti formalism for stochastic chemical kinetics is
proposed. Using the extension, path-integral expressions consistent with
previous studies are obtained. In addition, the extended formalism is naturally
connected to orthogonal polynomials. We show that two different orthogonal
polynomials, i.e., Charlier polynomials and Hermite polynomials, can be used to
express the Doi-Peliti formalism explicitly.Comment: 10 page
Topological quantum phase transition in the BEC-BCS crossover phenomena
A crossover between the Bose Einstein condensation (BEC) and BCS
superconducting state is described topologically in the chiral symmetric
fermion system with attractive interaction. Using a local Z_2 Berry phase, we
found a quantum phase transition between the BEC and BCS phases without
accompanying the bulk gap closing.Comment: 4 pages, 5 figure
Pressure formulas for liquid metals and plasmas based on the density-functional theory
At first, pressure formulas for the electrons under the external potential
produced by fixed nuclei are derived both in the surface integral and volume
integral forms concerning an arbitrary volume chosen in the system; the surface
integral form is described by a pressure tensor consisting of a sum of the
kinetic and exchange-correlation parts in the density-functional theory, and
the volume integral form represents the virial theorem with subtraction of the
nuclear virial. Secondly on the basis of these formulas, the thermodynamical
pressure of liquid metals and plasmas is represented in the forms of the
surface integral and the volume integral including the nuclear contribution.
From these results, we obtain a virial pressure formula for liquid metals,
which is more accurate and simpler than the standard representation. From the
view point of our formulation, some comments are made on pressure formulas
derived previously and on a definition of pressure widely used.Comment: 18 pages, no figur
Equilibration in long-range quantum spin systems from a BBGKY perspective
The time evolution of -spin reduced density operators is studied for a
class of Heisenberg-type quantum spin models with long-range interactions. In
the framework of the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY)
hierarchy, we introduce an unconventional representation, different from the
usual cluster expansion, which casts the hierarchy into the form of a
second-order recursion. This structure suggests a scaling of the expansion
coefficients and the corresponding time scales in powers of with the
system size , implying a separation of time scales in the large system
limit. For special parameter values and initial conditions, we can show
analytically that closing the BBGKY hierarchy by neglecting -spin
correlations does never lead to equilibration, but gives rise to quasi-periodic
time evolution with at most independent frequencies. Moreover, for the
same special parameter values and in the large- limit, we solve the complete
recursion relation (the full BBGKY hierarchy), observing a superexponential
decay to equilibrium in rescaled time .Comment: 3 figure
Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
The Anderson model for independent electrons in a disordered potential is
transformed analytically and exactly to a basis of random extended states
leading to a variant of augmented space. In addition to the widely-accepted
phase diagrams in all physical dimensions, a plethora of additional, weaker
Anderson transitions are found, characterized by the long-distance behavior of
states. Critical disorders are found for Anderson transitions at which the
asymptotically dominant sector of augmented space changes for all states at the
same disorder. At fixed disorder, critical energies are also found at which the
localization properties of states are singular. Under the approximation of
single-parameter scaling, this phase diagram reduces to the widely-accepted one
in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson
transition at infinitesimal disorder, there is a transition between two
localized states, characterized by a change in the nature of wave function
decay.Comment: 51 pages including 4 figures, revised 30 November 200
Polynomial solutions of nonlinear integral equations
We analyze the polynomial solutions of a nonlinear integral equation,
generalizing the work of C. Bender and E. Ben-Naim. We show that, in some
cases, an orthogonal solution exists and we give its general form in terms of
kernel polynomials.Comment: 10 page
How to construct spin chains with perfect state transfer
It is shown how to systematically construct the quantum spin chains with
nearest-neighbor interactions that allow perfect state transfer (PST). Sets of
orthogonal polynomials (OPs) are in correspondence with such systems. The key
observation is that for any admissible one-excitation energy spectrum, the
weight function of the associated OPs is uniquely prescribed. This entails the
complete characterization of these PST models with the mirror symmetry property
arising as a corollary. A simple and efficient algorithm to obtain the
corresponding Hamiltonians is presented. A new model connected to a special
case of the symmetric -Racah polynomials is offered. It is also explained
how additional models with PST can be derived from a parent system by removing
energy levels from the one-excitation spectrum of the latter. This is achieved
through Christoffel transformations and is also completely constructive in
regards to the Hamiltonians.Comment: 7 page
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