35 research outputs found
On polynomial solutions of Heun equation
By making use of a recently developed method to solve linear differential
equations of arbitrary order, we find a wide class of polynomial solutions to
the Heun equation. We construct the series solution to the Heun equation before
identifying the polynomial solutions. The Heun equation extended by the
addition of a term, - \s/x, is also amenable for polynomial solutions.Comment: 4 pages, No figur
Coherent states of P{\"o}schl-Teller potential and their revival dynamics
A recently developed algebraic approach for constructing coherent states for
solvable potentials is used to obtain the displacement operator coherent state
of the P\"{o}schl-Teller potential. We establish the connection between this
and the annihilation operator coherent state and compare their properties. We
study the details of the revival structure arising from different time scales
underlying the quadratic energy spectrum of this system.Comment: 13 pages, 6 figure
A Unified Algebraic Approach to Few and Many-Body Correlated Systems
The present article is an extended version of the paper {\it Phys. Rev.} {\bf
B 59}, R2490 (1999), where, we have established the equivalence of the
Calogero-Sutherland model to decoupled oscillators. Here, we first employ the
same approach for finding the eigenstates of a large class of Hamiltonians,
dealing with correlated systems. A number of few and many-body interacting
models are studied and the relationship between their respective Hilbert
spaces, with that of oscillators, is found. This connection is then used to
obtain the spectrum generating algebras for these systems and make an algebraic
statement about correlated systems. The procedure to generate new solvable
interacting models is outlined. We then point out the inadequacies of the
present technique and make use of a novel method for solving linear
differential equations to diagonalize the Sutherland model and establish a
precise connection between this correlated system's wave functions, with those
of the free particles on a circle. In the process, we obtain a new expression
for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having
Laughlin wave function as the ground-state and point out the natural emergence
of the underlying linear symmetry in this approach.Comment: 18 pages, Revtex format, To appear in Physical Review
On realizations of nonlinear Lie algebras by differential operators
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra
in terms of differential operators strongly related to bosonic operators. We
also distinguish their finite- and infinite-dimensional representations. The
linear, quadratic and cubic cases are explicitly visited but the method works
for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are
studied in the context of these ``nonlinear'' Lie algebras and some examples
dealing with quantum optics are pointed out.Comment: 21 pages, Latex; New examples added in Sect.
Quantum Calogero-Moser Models: Integrability for all Root Systems
The issues related to the integrability of quantum Calogero-Moser models
based on any root systems are addressed. For the models with degenerate
potentials, i.e. the rational with/without the harmonic confining force, the
hyperbolic and the trigonometric, we demonstrate the following for all the root
systems: (i) Construction of a complete set of quantum conserved quantities in
terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal
R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the
Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of
the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack
polynomials are defined for all root systems as unique eigenfunctions of the
Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v)
Algebraic construction of all excited states in terms of creation operators.
These are mainly generalisations of the results known for the models based on
the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure
Duality and quasiparticles in the Calogero-Sutherland model: Some exact results
The quantum-mechanical many-body system with the potential proportional to
the pairwise inverse-square distance possesses a strong-weak coupling duality.
Based on this duality, particle and/or quasiparticle states are described as
SU(1,1) coherent states. The constructed quasiparticle states are of
hierarchical nature.Comment: RevTeX, 10 page
Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators
A similarity transformation is constructed through which a system of
particles interacting with inverse-square two-body and harmonic potentials in
one dimension, can be mapped identically, to a set of free harmonic
oscillators. This equivalence provides a straightforward method to find the
complete set of eigenfunctions, the exact constants of motion and a linear
algebra associated with this model. It is also demonstrated that
a large class of models with long-range interactions, both in one and higher
dimensions can be made equivalent to decoupled oscillators.Comment: 9 pages, REVTeX, Completely revised, few new equations and references
are adde
Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
The development in the study of supersymmetric many-particle quantum systems
with inverse-square interactions is reviewed. The main emphasis is on quantum
systems with dynamical OSp(2|2) supersymmetry. Several results related to
exactly solved supersymmetric rational Calogero model, including shape
invariance, equivalence to a system of free superoscillators and non-uniqueness
in the construction of the Hamiltonian, are presented in some detail. This
review also includes a formulation of pseudo-hermitian supersymmetric quantum
systems with a special emphasis on rational Calogero model. There are quite a
few number of many-particle quantum systems with inverse-square interactions
which are not exactly solved for a complete set of states in spite of the
construction of infinitely many exact eigen functions and eigenvalues. The
Calogero-Marchioro model with dynamical SU(1,1|2) supersymmetry and a quantum
system related to short-range Dyson model belong to this class and certain
aspects of these models are reviewed. Several other related and important
developments are briefly summarized.Comment: LateX, 65 pages, Added Acknowledgment, Discussions and References,
Version to appear in Jouranl of Physics A: Mathematical and Theoretical
(Commissioned Topical Review Article
Algebra of the observables in the Calogero model and in the Chern-Simons matrix model
The algebra of observables of an N-body Calogero model is represented on the
S_N-symmetric subspace of the positive definite Fock space. We discuss some
general properties of the algebra and construct four different realizations of
the dynamical symmetry algebra of the Calogero model. Using the fact that the
minimal algebra of observables is common to the Calogero model and the finite
Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model.
We point out the algebraic similarities and distinctions of these models.Comment: 24 pages, misprints corrected, reference added, final version, to
appear in PR
Inequivalent quantization of the rational Calogero model with a Coulomb type interaction
We consider the inequivalent quantizations of a -body rational Calogero
model with a Coulomb type interaction. It is shown that for certain range of
the coupling constants, this system admits a one-parameter family of
self-adjoint extensions. We analyze both the bound and scattering state sectors
and find novel solutions of this model. We also find the ladder operators for
this system, with which the previously known solutions can be constructed.Comment: 15 pages, 3 figures, revtex4, typos corrected, to appear in EPJ