32 research outputs found
Multi-indexed (q-)Racah Polynomials
As the second stage of the project multi-indexed orthogonal polynomials, we
present, in the framework of `discrete quantum mechanics' with real shifts in
one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from
the (q-)Racah polynomials by multiple application of the discrete analogue of
the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state'
vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials
reported earlier. The virtual state vectors are the `solutions' of the matrix
Schr\"odinger equation with negative `eigenvalues', except for one of the two
boundary points.Comment: 29 pages. The type II (q-)Racah polynomials are deleted because they
can be obtained from the type I polynomials. To appear in J.Phys.
Constrained Dynamics of an Anomalous Relativistic Spinning Particle in Electromagnetic Background
In this paper we have considered the dynamics of an anomalous ()
charged relativistic spinning particle in the presence of an external
electromagnetic field. The constraint analysis is done and the complete set of
Dirac brackets are provided that generate the canonical Lorentz algebra and
dynamics through Hamiltonian equations of motion. The spin-induced effective
curvature of spacetime and its possible connection with Analogue Gravity models
are commented upon.Comment: 10 pages Latex, minor corrections and changes in ref., slightly
enlarged version, to appear in EPJ
From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics
In 1990, Dyson published a proof due to Feynman of the Maxwell equations
assuming only the commutation relations between position and velocity. With
this minimal assumption, Feynman never supposed the existence of Hamiltonian or
Lagrangian formalism. In the present communication, we review the study of a
relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's
magnetic angular momentum and Dirac magnetic monopole are the consequences of
the structure of the Lorentz Lie algebra defined by the Feynman's brackets.
Then, we extend these ideas to the dual momentum space by considering
noncommutative quantum mechanics. In this context, we show that the
noncommutativity of the coordinates is responsible for a new effect called the
spin Hall effect. We also show its relation with the Berry phase notion. As a
practical application, we found an unusual spin-orbit contribution of a
nonrelativistic particle that could be experimentally tested. Another practical
application is the Berry phase effect on the propagation of light in
inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland,
U.S.A., August 2006
On the linear forms of the Schrodinger equation
Generalizing the linearisation procedure used by Dirac and later by
L\'evy-Leblond, we derive the first-order non-relativistic wave equations for
particles of spin 1 and spin 3/2 starting from the Schrodinger equation
Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry
Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian
Hamiltonian H_{-}=\omega(\xi^{\dag} \xi+\1/2)+\alpha \xi^{2}+\beta \xi^{\dag
2}, where and is a first order differential
operator, to obtain the partner potentials and which are
new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian
equivalents of the non-Hermitian partner Hamiltonians . We have
provided an algebraic way to obtain the spectrum and wavefunctions of a
nonlinear isotonic oscillator. The solutions of which are Hermitian
counterparts of Swanson Hamiltonian are obtained under some parameter
restrictions that are found. Also, we have checked that if the intertwining
operator satisfies , where and is the first order differential operator,
which factorizes Hermitian equivalents of .Comment: 11 page
Disconjugacy, regularity of multi-indexed rationally extended potentials, and Laguerre exceptional polynomials
Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
In recent years, one of the most interesting developments in quantum
mechanics has been the construction of new exactly solvable potentials
connected with the appearance of families of exceptional orthogonal polynomials
(EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre
and Hermite) classical orthogonal polynomials, which start with a constant, the
EOP families begin with some polynomial of degree greater than or equal to one,
but still form complete, orthogonal sets with respect to some positive-definite
measure. We show how they may appear in the bound-state wavefunctions of some
rational extensions of well-known exactly solvable quantum potentials. Such
rational extensions are most easily constructed in the framework of
supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class
of translationally shape invariant potentials. We review the most recent
results in this field, which use higher-order SUSYQM. We also comment on some
recent re-examinations of the shape invariance condition, which are independent
of the EOP construction problem.Comment: 21 pages, no figure; communication at the Symposium Symmetries in
Science XV, July 31-August 5, 2011, Bregenz, Austri
One-Dimensional Fermi liquids
I attempt to give a pedagogical overview of the progress which has occurred
during the past decade in the description of one-dimensional correlated
fermions. Fermi liquid theory based on a quasi-particle picture, breaks down in
one dimension because of the Peierls divergence and because of charge-spin
separation. It is replaced by a Luttinger liquid whose elementary excitations
are collective charge and spin modes, based on the exactly solvable Luttinger
model. I review this model and various solutions with emphasis on bosonization
(and its equivalence to conformal field theory), and its physical properties.
The notion of a Luttinger liquid implies that all gapless 1D systems share
these properties at low energies.
Chapters 1 and 2 of the article contain an introduction and a discussion of
the breakdown of Fermi liquid theory. Chapter 3 describes in detail the
solution of the Luttinger model both by bosonization and by Green's functions
methods and summarizes the properties of the model, expressed thorugh
correlation functions. The relation to conformal field theory is discussed.
Chapter 4 of the article introduces the notion of a Luttinger liquid. It
describes in much detail the various mappings applied to realistic models of 1D
correlated fermions, onto the Luttinger model, as well as important corrections
to the Luttinger model properties discussed in Ch.3. Chapter 5 describes
situations where the Luttinger liquid is not a stable fixed point, and where
spin or charge gaps open in at least one channel. Chapter 6 discusses
multi-band and multichain problems, in particular the stability of a Luttinger
liquid with respect to interchain hopping. Ch. 7 gives a brief summary of
experimental efforts to uncover Luttinger liquid correlations in quasi-1D
materials.Comment: uuencoded Latex files and postscript figures, one Readme-file approx
160 pages + 13 figures; to be published by Reports on Progress in Physic
Extended Krein-Adler theorem for the translationally shape invariant potentials
Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Backlund transformations. It allows us to establish novel bi-linear Wronskian and determinantal identities for classical orthogonal polynomials