239 research outputs found
An Exactly Solvable Spin Chain Related to Hahn Polynomials
We study a linear spin chain which was originally introduced by Shi et al.
[Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength
contains a parameter and depends on the parity of the chain site.
Extending the model by a second parameter , it is shown that the single
fermion eigenstates of the Hamiltonian can be computed in explicit form. The
components of these eigenvectors turn out to be Hahn polynomials with
parameters and . The construction of the
eigenvectors relies on two new difference equations for Hahn polynomials. The
explicit knowledge of the eigenstates leads to a closed form expression for the
correlation function of the spin chain. We also discuss some aspects of a
-extension of this model
A classification of generalized quantum statistics associated with basic classical Lie superalgebras
Generalized quantum statistics such as para-statistics is usually
characterized by certain triple relations. In the case of para-Fermi statistics
these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1);
in the case of para-Bose statistics they are associated with the Lie
superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition
of ``a generalized quantum statistics associated with a classical Lie algebra
G'' was given, and a complete classification was obtained. Here, we consider
the definition of ``a generalized quantum statistics associated with a basic
classical Lie superalgebra G''. Just as in the Lie algebra case, this
definition is closely related to a certain Z-grading of G. We give in this
paper a complete classification of all generalized quantum statistics
associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and
D(m|n)
Solutions of the compatibility conditions for a Wigner quantum oscillator
We consider the compatibility conditions for a N-particle D-dimensional
Wigner quantum oscillator. These conditions can be rewritten as certain triple
relations involving anticommutators, so it is natural to look for solutions in
terms of Lie superalgebras. In the recent classification of ``generalized
quantum statistics'' for the basic classical Lie superalgebras
[math-ph/0504013], each such statistics is characterized by a set of creation
and annihilation operators plus a set of triple relations. In the present
letter, we investigate which cases of this classification also lead to
solutions of the compatibility conditions. Our analysis yields some known
solutions and several classes of new solutions.Comment: 9 page
Casimir invariants and characteristic identities for
A full set of (higher order) Casimir invariants for the Lie algebra
is constructed and shown to be well defined in the category
generated by the highest weight (unitarizable) irreducible
representations with only a finite number of non-zero weight components.
Moreover the eigenvalues of these Casimir invariants are determined explicitly
in terms of the highest weight. Characteristic identities satisfied by certain
(infinite) matrices with entries from are also determined and
generalize those previously obtained for by Bracken and Green.Comment: 10 pages, PlainTe
Deformed su(1,1) Algebra as a Model for Quantum Oscillators
The Lie algebra can be deformed by a reflection
operator, in such a way that the positive discrete series representations of
can be extended to representations of this deformed
algebra . Just as the positive discrete series
representations of can be used to model a quantum
oscillator with Meixner-Pollaczek polynomials as wave functions, the
corresponding representations of can be utilized to
construct models of a quantum oscillator. In this case, the wave functions are
expressed in terms of continuous dual Hahn polynomials. We study some
properties of these wave functions, and illustrate some features in plots. We
also discuss some interesting limits and special cases of the obtained
oscillator models
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