Luc Vinet

Exact operator solution of the Calogero-Sutherland model

The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.

Group actions on principal bundles and invariance conditions for gauge fields

Invariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification of group actions on bundles as automorphisms projecting to an action on a base manifold with a sufficiently regular orbit structure is given in terms of group homorphisms and a generalization of Wang’s theorem classifying invariant connections is derived. Illustrative examples on compactified Minkowski space are given.

Supersymmetry of the Pauli Equation in the Presence of a Magnetic Monopole

Abstract It is shown that the Pauli hamiltonian for a spin 1 2 particle in the presence of a Dirac magnetic monopole possesses a dynamical conformal OSp(1,1) supersymmetry. Using this symmetry, the spectrum is constructed explicitly, and all but the lowest angular momentum states transform under irreducible representations of the supergroup. The lowest angular momentum states transform under irreducible representations of the SO(2,1) - subgroup of OSp(1,1) only, because the supercharges are not self-adjoint in that sector.

LIE GROUP FORMALISM FOR DIFFERENCE EQUATIONS

The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples are analysed, one of them being a nonlinear difference equation. For the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit.

More on the q-oscillator algebra and q-orthogonal polynomials

Properties of certain q-orthogonal polynomials are connected to the q-oscillator algebra. The Wall and q-Laguerre polynomials are shown to arise as matrix elements of q-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous q-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 q-Hermite and the continuous q-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are derived algebraically.

Difference Schrödinger operators with linear and exponential discrete spectra

Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.

Lie symmetries of finite‐difference equations

Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations are determined and it is found that they retain the same Lie symmetry algebras as their continuum limits. Solutions with definite transformation properties are obtained; identities and formulas for these functions are then derived using the symmetry algebra.

Spectral transformations, self-similar reductions and orthogonal polynomials

We study spectral transformations in the theory of orthogonal polynomials which are similar to Darboux transformations for the Schr?dinger equation. Linear transformations of the Stieltjes function with coefficients that are rational in the argument are constructed as iterations of the Christoffel and Geronimus transformations. We describe a characteristic property of semi-classical orthogonal polynomials (SCOP) on the uniform and the exponential lattice; namely, that all these polynomials can be obtained through simple quasi-periodic and q-periodic (self-similar) closures of the chain of linear spectral transformations. In the self-similar setting, a characterization of the Laguerre - Hahn polynomials on linear and q-linear lattices is obtained by considering rational transformations of the Stieltjes function generated by transitions to the associated polynomials.

On the defining relations of quantum superalgebras

In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for slq(m, n) and ospq(m, 2n) usingq-oscillator representations.

Dynamical supersymmetry of the magnetic monopole and the

We examine the recently discovered dynamical OSp(1, 1) supersymmetry of the Pauli Hamiltonian for a spin 1/2 particle with gyromagnetic ratio 2, in the presence of a Dirac magnetic monopole. Using this symmetry and algebraic methods only, we construct the spectrum and obtain the wave functions. At all but the lowest angular momenta, the states transform under a single irreducible representation of OSp(1, 1). On the lowest angular momentum states, it is impossible to define self-adjoint supercharges, and the states transform under an irreducible representation of SO(2, 1) only. The Hamiltonian is not self-adjoint in thes-wave sector, but admits a one parameter family of self-adjoint extensions. The full SO(2, 1) algebra can be realized only for two specific values of the parameter.The Pauli Hamiltonian is generalized to accommodate aλ2/r2 potential. The new Hamiltonian exhibits a dynamical OSp(2, 1) supersymmetry. The spectrum and the wave functions are obtained. The states at all but the lowest angular momenta transform under the sum of two irreducible representations of OSp(2, 1). These two representations are distinguished by the “chirality” of their ground state. On the lowest angular momentum states, the OSp(2, 1) group is still realized, since the supercharges can all be rendered self-adjoint simultaneously, but the states only transform according to a single irreducible representation of OSp(2, 1). The chirality of the ground state for this representation is related to the signs ofλ andeg. The Hamiltonian is not self-adjoint in thes-wave sector when |λ|<3/2. Only one of its self-adjoint extensions supports the OSp(2, 1) supersymmetry, and yields the wave functions obtained from the group theoretic approach. The supersymmetry is always spontaneously broken as there exists no normalizable zero energy states.The massless Dirac Hamiltonian in the presence of a magnetic monopole and aλ/r potential is related to a generator of an OSp(2, 1) superalgebra which also contains the Pauli Hamiltonian. This symmetry is used to generate the complete spectrum of the Dirac Hamiltonian.