Jacques Deenen

Partially coherent states of the real symplectic group

In the present paper, we introduce partially coherent states for the positive discrete series irreducible representations 〈λd+n/2,...,λ1+n/2〉 of Sp(2d,R), encountered in physical applications. These states are characterized by both continuous and discrete labels. The latter specify the row of the irreducible representation [λ1λ2⋅⋅⋅λd] of the maximal compact subgroup U(d), while the former parametrize an element of the factor space Sp(2d,R)/H, where H is the Sp(2d,R) subgroup leaving the [λ1λ2⋅⋅⋅λd] representation space invariant. We consider three classes of partially coherent states, respectively, generalizing the Perelomov and Barut–Girardello coherent states, as well as some recently introduced intermediate coherent states. We prove that each family of partially coherent states forms an overcomplete set in the representation space of 〈λd+n/2,...,λ1+n/2〉, and study its generating function properties. We show that it leads to a representation of the Sp(2d,R) generators in the form of differential operato...

Dynamical group of microscopic collective states. I. One‐dimensional case

In the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an A‐nucleon system, defined as those A‐particle states invariant under the orthogonal group O(n) associated with the n = A−1 Jacobi vectors. The present paper discusses the case of a hypothetical one‐dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group Spc(2,R), which is the restriction to the collective subspace of the group Sp(2,R) of linear canonical transformations in n dimensions conserving the O(n) symmetry. In addition to the well‐known realization of the dynamical group in the Schrodinger representation based upon the Dzublik–Zickendraht transformation, two new realizations are proposed. The first acts in a Barut Hilbert space, which is the subspace of a Bargmann Hilbert space of analytic functions left invariant by O(n). A unitary mapping is established between the ordi...

Dynamical group of microscopic collective states. III. Coherent state representations in d dimensions

The present series of papers deals with various realizations of the dynamical group Spc(2d,R) of microscopic collective states for an A nucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=A−1 relative Jacobi vectors. In this paper, we further study the Barut–Girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to Spc(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to Holstein–Primakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their gener...

Dynamical group of microscopic collective states. II. Boson representations in d dimensions

The present series of papers deals with various realizations of the dynamical group Spc(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of Spc(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space Fc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in Fc, classified according to the chain Spc(2d,R)⊇Uc(d), can be put into one‐to‐one correspondence with basis functions, classified according to the chain U (ν)⊇U (d), in a Bargmann Hilbert space B of analytic functions in ν complex variables representing ν‐dimensional boson states. By equating the complex variables of Fc and their conjugate momenta with th...

Boson representations of the real symplectic group and their applications to the nuclear collective model

Both non‐Hermitian Dyson and Hermitian Holstein–Primakoff representations of the Sp(2d,R) algebra are obtained when the latter is restricted to a positive discrete series irreducible representation 〈λd +n/2,...,λ1+n/2〉. For such purposes, some results for boson representations, recently deduced from a study of the Sp(2d,R) partially coherent states, are combined with some standard techniques of boson expansion theories. The introduction of Usui operators enables the establishment of useful relations between the various boson representations. Two Dyson representations of the Sp(2d,R) algebra are obtained in compact form in terms of ν=d(d+1)/2 pairs of boson creation and annihilation operators, and of an extra U(d) spin, characterized by the irreducible representation [λ1⋅⋅⋅λd]. In contrast to what happens when λ1=⋅⋅⋅=λd=λ, it is shown that the Holstein–Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation. Explicit expansions are,...

Canonical solution of the state labelling problem for SU(n)⊃SO(n) and Littlewood's branching rule. I. General formulation

The internal state labelling problem for the d-row irreducible representations of SU(n) (where 2d<or=n), when reduced with respect to SO(n), is shown to amount to the external state labelling problem for U(d). The canonical solution of the latter due to Biedenharn et al. (1967) provides a canonical solution of the former, which reflects the operation of Littlewoods branching rule for U(n) contains/implies O(n) in a very simple way.

Parity dependence in heavy-ion scattering

Abstract The parity dependence of collisions between p- and sd-shell nuclei is studied microscopically in the two-centre harmonic oscillator model. A simple rule giving the sign of the odd-even part of the potential is established. Besides the well-known dependence on the nucleon-number difference, a marked variation of the parity effect with the shell structure of the colliding nuclei is found and discussed. The odd-even dependence is shown to start decreasing beyond a critical angular momentum. The influence of choosing more realistic individual wave functions in the two-centre model is investigated in an approximate way.

Generators for nonlinear canonical transformations

Canonical transformations whose generators are linear in the basic operators Pj and arbitrary in the canonically conjugate operators Qj are explicitly constructed. It is shown that they correspond to gauge transformations and changes of variables. Some applications are mentioned in the one-dimensional case.

Determination of the Sp(2d,R) generator matrix elements through a boson mapping

For the discrete series irreducible representations (( lambda +n/2)d) of Sp(2d,R), the determination of the Sp(2d,R) generator matrix elements in an Sp(2d,R) contains/implies U(d) basis is reduced to the much simpler calculation of boson operator matrix elements between U( nu ) contains/implies U(d) boson states, where nu =d(d+1)/2. The key of this reduction is the previously derived Holstein-Primakoff boson representation of the Sp(2d,R) generators. As an illustration, the case of Sp(6,R) is worked out in detail.

Solvable potentials generated by SL(2, R)

Canonical transformations are used to build realisations of SL(2, R) in terms of the basic quantum mechanical operators Q and P. The results are used to construct solvable potentials in the framework of one-dimensional quantum mechanics.