Michèle Irac-Astaud

MOLECULAR-COHERENT-STATES AND MOLECULAR-FUNDAMENTAL-STATES

New families of Molecular-Coherent-States are constructed by the Perelomov group-method. Each family is generated by a Molecular-Fundamental-State that depends on an arbitrary sequence of complex numbers cj. Two of these families were already obtained by D. Janssen and by J. A. Morales, E. Deumens and Y. Ohrn. The properties of these families are investigated and we show that most of them are independent on the cj.

DIFFERENTIAL CALCULUS ON A THREE-PARAMETER OSCILLATOR ALGEBRA

Two differential calculi are developed on an algebra generalizing the usual q-oscillator algebra and involving three generators and three parameters. They are shown to be invariant under the same quantum group that is extended to a ten-generator Hopf algebra. We discuss the special case where it reduces to a deformation of the invariance group of the Weyl-Heisenberg algebra for which we prove the existence of a constraint between the values of the parameters.

A recursive linearization process for evolution equations of algebraic functionals

Abstract We construct an algebra of algebraic functionals and discuss some of its properties. We develop a recursive linearization process for first order differential evolution equations in this mathematical frame.

A new approach to perturbation theory: star diagrams

Perturbation theory for first‐order nonlinear differential equations with source is developed in a new way, and associated with diagrams that we call star diagrams. In some cases the method allows one to express the n‐point functions in explicit form.

Bargmann Representation for Some Deformed Harmonic Oscillators with Non-Fock Representation

We prove that Bargmann representations exist for some deformed harmonic oscillators that admit non-Fock representations. In specific cases, we explicitly obtain the resolution of the identity in terms of a true integral on the complex plane. We prove in explicit examples that Bargmann representations cannot always be found, particularly when the coherent states do not exist in the whole complex plane.

TheSU(1,1) coherent states related to the affine group wavelets

The Perelomov coherent states ofSU(1,1) are labeled by elements of the quotient ofSU(1,1) by its rotation subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group. Such a formulation permits to find new properties of theSU(1,1) coherent states and to relate them to affine wavelets.

From Bargmann representations to Deformed Harmonic Oscillator Algebras

Deformed Harmonic Oscillator Algebras (DHOA) are generated by four operators: two mutually adjoint a and a†, self-adjoint N, and the unity 1. The Bargmann-Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of DHOA act on a Bargmann-Hilbert space, and the creation (or the annihilation operator) is the multiplication by z. We discuss conditions for the existence of DHOA assumed to admit a given Bargmann representation.

Unitary representations of the quantum algebra suq(2) on a real two-dimensional sphere for q∈R+ or generic q∈S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra suq(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions, and are related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials. In their study, the values of q were implicitly restricted to q∈R+. In the present paper, we extend their work to the case of generic values of q∈S1 (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q∈R+ and generic q∈S1, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.

Bargmann Representations for Deformed Harmonic Oscillators

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a†, N and the unity 1 such as [a,N]=a,[a†,N]=-a†,a†a=ψ(N) and aa†=ψ(N+1). We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of a (or a†). We give various examples, in particular we consider functions ψ that are linear combinations of qN, q-N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.

Unitary representations of suq(2) on the plane for q ∈ R+ or generic q ∈ S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra suq(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q ∈ R+. In the present paper, we extend their work to the case of generic values of q ∈ S1 (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q ∈ R+ and generic q ∈ S1, by determining some appropriate scalar products.