1,397 research outputs found
A Calabi-Yau algebra with symmetry and the Clebsch-Gordan series of
Building on classical invariant theory, it is observed that the polarised
traces generate the centraliser of the diagonal embedding of
in . The paper then focuses on and the
case . A Calabi--Yau algebra with three generators is
introduced and explicitly shown to possess a PBW basis and a certain central
element. It is seen that is isomorphic to a quotient of the
algebra by a single explicit relation fixing the value of the
central element. Upon concentrating on three highest weight representations
occurring in the Clebsch--Gordan series of , a specialisation of
arises, involving the pairs of numbers characterising the three
highest weights. In this realisation in , the
coefficients in the defining relations and the value of the central element
have degrees that correspond to the fundamental degrees of the Weyl group of
type . With the correct association between the six parameters of the
representations and some roots of , the symmetry under the full Weyl group
of type is made manifest. The coefficients of the relations and the value
of the central element in the realisation in are
expressed in terms of the fundamental invariant polynomials associated to
. It is also shown that the relations of the algebra can be
realised with Heun type operators in the Racah or Hahn algebra.Comment: 24 page
Generalized squeezed-coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality
A set of generalized squeezed-coherent states for the finite u(2) oscillator
is obtained. These states are given as linear combinations of the mode
eigenstates with amplitudes determined by matrix elements of exponentials in
the su(2) generators. These matrix elements are given in the (N+1)-dimensional
basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix
multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the
Krawtchouk and vector-orthogonal polynomials as their building blocks. The
algebraic setting allows for the characterization of these polynomials and the
computation of mean values in the squeezed-coherent states. In the limit where
N goes to infinity and the discrete oscillator approaches the standard harmonic
oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the
squeezed-coherent states tend to those of the standard oscillator.Comment: 18 pages, 1 figur
An infinite family of superintegrable Hamiltonians with reflection in the plane
We introduce a new infinite class of superintegrable quantum systems in the
plane. Their Hamiltonians involve reflection operators. The associated
Schr\"odinger equations admit separation of variables in polar coordinates and
are exactly solvable. The angular part of the wave function is expressed in
terms of little -1 Jacobi polynomials. The spectra exhibit "accidental"
degeneracies. The superintegrability of the model is proved using the
recurrence relation approach. The (higher-order) constants of motion are
constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page
The missing label of and its symmetry
We present explicit formulas for the operators providing missing labels for
the tensor product of two irreducible representations of . The
result is seen as a particular representation of the diagonal centraliser of
through a pair of tridiagonal matrices. Using these explicit
formulas, we investigate the symmetry of this missing label problem and we find
a symmetry group of order 144 larger than what can be expected from the natural
symmetries. Several realisations of this symmetry group are given, including an
interpretation as a subgroup of the Weyl group of type , which appeared in
an earlier work as the symmetry group of the diagonal centraliser. Using the
combinatorics of the root system of type , we provide a family of
representations of the diagonal centraliser by infinite tridiagonal matrices,
from which all the finite-dimensional representations affording the missing
label can be extracted. Besides, some connections with the Hahn algebra,
Heun--Hahn operators and Bethe ansatz are discussed along with some
similarities with the well-known symmetries of the Clebsch--Gordan
coefficients
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
Superspace realizations of the Bannai-Ito algebra
A model of the Bannai-Ito algebra in a superspace is introduced. It is
obtained from the three-fold tensor product of the basic realization of the Lie
superalgebra in terms of operators in one continuous and one
Grassmanian variable. The basis vectors of the resulting Bannai-Ito algebra
module involve Jacobi polynomials
A superintegrable finite oscillator in two dimensions with SU(2) symmetry
A superintegrable finite model of the quantum isotropic oscillator in two
dimensions is introduced. It is defined on a uniform lattice of triangular
shape. The constants of the motion for the model form an SU(2) symmetry
algebra. It is found that the dynamical difference eigenvalue equation can be
written in terms of creation and annihilation operators. The wavefunctions of
the Hamiltonian are expressed in terms of two known families of bivariate
Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials
form bases for SU(2) irreducible representations. It is further shown that the
pair of eigenvalue equations for each of these families are related to each
other by an SU(2) automorphism. A finite model of the anisotropic oscillator
that has wavefunctions expressed in terms of the same Rahman polynomials is
also introduced. In the continuum limit, when the number of grid points goes to
infinity, standard two-dimensional harmonic oscillators are obtained. The
analysis provides the limit of the bivariate Krawtchouk
polynomials as a product of one-variable Hermite polynomials
Supersymmetric Quantum Mechanics with Reflections
We consider a realization of supersymmetric quantum mechanics where
supercharges are differential-difference operators with reflections. A
supersymmetric system with an extended Scarf I potential is presented and
analyzed. Its eigenfunctions are given in terms of little -1 Jacobi polynomials
which obey an eigenvalue equation of Dunkl type and arise as a q-> -1 limit of
the little q-Jacobi polynomials. Intertwining operators connecting the wave
functions of extended Scarf I potentials with different parameters are
presented.Comment: 17 page
More on the q-oscillator algebra and q-orthogonal polynomials
Properties of certain -orthogonal polynomials are connected to the
-oscillator algebra. The Wall and -Laguerre polynomials are shown to
arise as matrix elements of -exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big -Hermite polynomials and the continuous
-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived
Anomalous density of states in a metallic film in proximity with a superconductor
We investigated the local electronic density of states in
superconductor-normal metal (Nb-Au) bilayers using a very low temperature (60
mK) STM. High resolution tunneling spectra measured on the normal metal (Au)
surface show a clear proximity effect with an energy gap of reduced amplitude
compared to the bulk superconductor (Nb) gap. Within this mini-gap, the density
of states does not reach zero and shows clear sub-gap features. We show that
the experimental spectra cannot be described with the well-established Usadel
equations from the quasi-classical theory.Comment: 4 pages, 5 figure
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