102 research outputs found
Equivalences of the Multi-Indexed Orthogonal Polynomials
Multi-indexed orthogonal polynomials describe eigenfunctions of exactly
solvable shape-invariant quantum mechanical systems in one dimension obtained
by the method of virtual states deletion. Multi-indexed orthogonal polynomials
are labeled by a set of degrees of polynomial parts of virtual state
wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi,
Wilson and Askey-Wilson types, two different index sets may give equivalent
multi-indexed orthogonal polynomials. We clarify these equivalences.
Multi-indexed orthogonal polynomials with both type I and II indices are
proportional to those of type I indices only (or type II indices only) with
shifted parameters.Comment: 25 pages. Some comments and a reference added. To appear in
J.Math.Phy
Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
We consider the exactly solvable quantum mechanical systems whose
eigenfunctions are described by the multi-indexed orthogonal polynomials of
Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the
recurrence relations with constant coefficients for the -indexed orthogonal
polynomials, it is expected that the systems satisfy the generalized closure
relations. In fact we can verify this statement for small examples. The
generalized closure relation gives the exact Heisenberg operator solution of a
certain operator, from which the creation and annihilation operators of the
system are obtained.Comment: 33 page
Unitary Representations of Infinity Algebras
We study the irreducible unitary highest weight representations, which are
obtained from free field realizations, of infinity algebras (,
, , , ,
) with central charges (, , , , , ).
The characters of these representations are computed. We construct a new
extended superalgebra , whose bosonic sector is
. Its representations obtained from a free
field realization with central charge , are classified into two classes:
continuous series and discrete series. For the former there exists a
supersymmetry, but for the latter a supersymmetry exists only for .Comment: 20 page
Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
In a previous paper we presented term recurrence relations with
variable dependent coefficients for -indexed orthogonal polynomials of
Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present
(conjectures of) the recurrence relations with constant coefficients for these
multi-indexed orthogonal polynomials. The simplest recurrence relations have
terms, where is the degree of the lowest member of
the multi-indexed orthogonal polynomials.Comment: 27 pages. Comments, references and examples added, reference
information updated. To appear in J.Math.Phy
Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
In a previous paper, we presented conjectures of the recurrence relations
with constant coefficients for the multi-indexed orthogonal polynomials of
Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a
proof for the Laguerre and Jacobi cases. Their bispectral properties are also
discussed, which give a method to obtain the coefficients of the recurrence
relations explicitly. This paper extends to the Laguerre and Jacobi cases the
bispectral techniques recently introduced by G\'omez-Ullate et al. to derive
explicit expressions for the coefficients of the recurrence relations satisfied
by exceptional polynomials of Hermite type.Comment: 37 pages. Comments added, typo in (A.15) corrected, reference
information updated. To appear in J.Math.Phy
Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
In our previous papers, the Wronskian identities for the Hermite, Laguerre
and Jacobi polynomials and the Casoratian identities for the Askey-Wilson
polynomial and its reduced form polynomials were presented. These identities
are naturally derived through quantum mechanical formulation of the classical
orthogonal polynomials; ordinary quantum mechanics for the former and discrete
quantum mechanics with pure imaginary shifts for the latter. In this paper we
present the corresponding identities for the discrete quantum mechanics with
real shifts. Infinitely many Casoratian identities for the -Racah polynomial
and its reduced form polynomials are obtained.Comment: 37 pages. Comments, a reference and proportionality constants for
q-Racah case are added. Sec.3.3 is moved to App.B. To appear in PTE
Exact Heisenberg operator solutions for multi-particle quantum mechanics
Exact Heisenberg operator solutions for independent `sinusoidal coordinates'
as many as the degree of freedom are derived for typical exactly solvable
multi-particle quantum mechanical systems, the Calogero systems based on any
root system. These Heisenberg operator solutions also present the explicit
forms of the annihilation-creation operators for various quanta in the
interacting multi-particle systems. At the same time they can be interpreted as
multi-variable generalisation of the three term recursion relations for
multi-variable orthogonal polynomials constituting the eigenfunctions.Comment: 17 pages, no figure
Extensions of solvable potentials with finitely many discrete eigenstates
We address the problem of rational extensions of six examples of
shape-invariant potentials having finitely many discrete eigenstates. The
overshoot eigenfunctions are proposed as candidates unique in this group for
the virtual state wavefunctions, which are an essential ingredient for
multi-indexed and iso-spectral extensions of these potentials. They have
exactly the same form as the eigenfunctions but their degrees are much higher
than n_{max} so that their energies are lower than the groundstate energy.Comment: 22 pages, 3 figures. Typo corrected, comments and references added.
To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1212.659
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