Unitary group decomposition of hamiltonian operators. II. SU (4) irreducible tensors,norms and their energy variation and symmetry breaking

SU(6) ? SU(4) tensor decomposition of effective interactions in the 2s-1d shell has been carried out to examine the relative importance of the various irreducible tensors in many-particle spaces. For this purpose norms of the irreducible tensors are evaluated in many-particle spaces. Variation of the expectation value of the square of the irreducible tensor parts with excitation energy has also been examined using the polynomial expansion method. A new measure of symmetry breaking that is theoretically more sound is derived which includes in its definition partial width as well as internal width. This is used to study SU(4) symmetry mixing in nuclei.     

Dynamical basis generation and structure of the Hartree-Fock approximation

A variational method is developed based on the Hartree-Fock approximation, but not restricted to a single Slater determinant trial space. The idea is to find a subspace of collective states which are strongly coupled to the ground state by providing a systematic technique to generate these basis states from a Hartree-Fock-like state. In the resulting basis space a residual diagonalization is easily performed. An application to a solvable model is made, both to justify and to investigate the structure of our approach.     

Commutators and self-adjointness of Hamiltonian operators

A time dependent approach to self-adjointness is presented and it is applied to quantum mechanical Hamiltonians which are not semi-bounded. Sufficient conditions are given for self-adjointness of Schrödinger and Dirac Hamiltonians with potentials which are unbounded at infinity. The method is the introduction of an auxiliary operatorN≧0 whose rate of change (commutator with the Hamiltonian) is bounded by a multiple ofN.     

Variational derivation of the adiabatic time-dependent Hartree-Fock formalism and generalized Hamiltonian dynamics

The three-particle collision process 3 → 3 with relative high-energy motion of each pair of particles described by a model with eikonal Hamiltonian is investigated. No additional restrictions on the motion of the particles (such as fixed scattering centre approximation) are imposed. It is shown that the three-particle problem in this case can be solved analytically. An explicit expression for 3 → 3 amplitude off the energy shell is obtained as the result of the exact summation of the multiple scattering series in the considered model. On the energy shell this series breaks off (there are no terms higher than triple). The formula for the mutual cancellation of the higher-order terms of the series is derived.     

Momentum and Hamiltonian operators in generalized coordinates

The self-adjointness of momentum operators in generalized coordinates, questioned by Domingos and Caldeira is shown. The momentum operators of a particle and the kinetic part of its Hamiltonian operator constructed from them are characterized as self-adjoint operators and geometrical objects in coordinate-free form. Local coordinates of ann-dimensional Riemannian manifold are taken as the generalized coordinates of the particle. As an example the curvilinear coordinates of Euclidean space are treated. The coefficients of connection and curvature are given on the manifold for which the assumed momentum operators exist. It is found that if our momentum operators form a complete set of mutually commuting observables, the manifold is locally Euclidean, i.e., there exists a local coordinate system such that we obtain the usual Schrödinger correspondence rule.     

A note on the truncation of the basis in Hartree-Fock calculation

Using different truncations of the basis, the Hartree-Fock calculation for 2s-1d shell nuclei with density dependent force has been performed. The influence of these truncations on the Hartree-Fock solutions has been analysed. Stability, shape and centre-of-mass motion energy of the nuclei studied are found to be mostly influenced.It is a pleasure to thank Dr. G. Ripka for many enlightening discussions. The computer calculations have been carried out during the stay at CEN Saclay, France.     

Recursion operators and Hamiltonian structures in Sato's theory

Following Sato's famous construction of the KP hierarchy as a hierarchy of commuting Lax equations on the algebra of microdifferential operators, it is shown that n-reduction leads to a recursive scheme for these equations. Explicit expressions for the recursion operators and the Hamiltonian operators are obtained.     

On the ascent of infinite dimensional Hamiltonian operators

In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 × 2 infinite dimensional Hamiltonian operator is 1 and the ascent of a class of 4 × 4 infinite dimensional Hamiltonian operators that arises in study of elasticity is2 are obtained. Concrete examples are given to illustrate the effectiveness of criterions.     

Effective-interaction expansions in the Hartree-Fock basis and effects of “back-door” intruders

The use of the Hartrree-Fock (HF) Hamiltonian in effective-interaction expansions is found to “push” out the branch point singularities which are located on the positive side of the origin in the complex x-plane. The effect may however be the opposite for “back-door” intruders, which correspond to such singularities in the negative half-plane. The dependence of these behaviors on the HF stability conditions and the sign of the particle-hole interactions is discussed.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Unitary and non-unitary representations of the complex inhomogeneous Lorentz group

We construct unitary and non-unitary representations of the complex inhomogeneous Lorentz group, including all its unitary, irreducible representations. We discuss the decomposition of these representations when they are restricted to the real inhomogeneous Lorentz group. We also discuss the representations of the Poincaré group for which the translation subgroup transforms under a not necessarily unitary representation. We summarize briefly the physical motivation for this study.Supported by the National Science Foundation (NSF-GP-58) while at Brandeis University and by the National Science Foundation, Grant NSF-GP-3465, while at the Courant Institute of Mathematical Sciences.     

Structure analysis of group I plant nucleases

Anticancer drugs attacking nucleic acids of the target cells have so far been based on animal or fungal ribonucleases. Plant nucleases have been proved to exhibit decreased cytotoxic side effects. Tomato bifunctional nuclease 1 with activity against both single-stranded and double-stranded RNA and DNA was produced in tobacco leaves as recombinant protein. The enzyme crystallizes under several different crystallization conditions. The presence of Zn(2+) ions was confirmed by X-ray fluorescence. First crystallographic data were obtained.     

Symmetry of the mixing Hamiltonian in lattices with a basis

To determine substitutional and injection superstructures in alloys having a complex lattice with a basis by the method of concentration waves it is necessary to know the eigenvectors of the Fourier transform matrices V_pq() of the mixing potential of a homogeneous solid solution. A technique has been developed, investigating the variations of the matrix elements V_pq() for symmetry transformations of the spatial lattice group, to solve this problem. The symmetry elements belonging to the wave vector group determine the internal structure of the matrices V_pq() for lattices with a basis. Symmetry elements not appearing in the wave vector group determine the relation between matrices belonging to different rays of the star wave vector. Restrictions on the matrix elements V_pq(), following from symmetry, are found for an h.c.p. lattice. The eigenvectors are calculated for the matrices V_pq(), whose vectors terminate in high-symmetry points, lines, and planes of the Brillouin zone. This made it possible to separate the problem of thermodynamic competition of intermediate phases in an alloy with a hexagonal lattice into symmetric and potential parts, similarly to the way this occurs in Bravais lattices.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 13–25, August, 1993.     

A new method of constructing symmetry-adapted linear combinations for a finite group: II. Correspondence theorem and basis operators

A correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry-adapted linear combination belonging to the same irreducible representation of a point group. By means of basis operators, the general expressions are given for the symmetry-adapted linear combinations, the coupling coefficients, the lattice harmonics and the hybrid atomic orbitals. Illustrative examples are discussed to demonstrate the effectiveness of the formalism.     

Recursion operators and bi-Hamiltonian structures in multidimensions. I

The algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied. In particular, the factorization of certain operators, called recursion operators, establishes the bi-Hamiltonian nature of all these equations. Recently, we have presented the recursion operator and the bi-Hamiltonian formulation of the Kadomtsev-Petviashvili equation, a two spatial dimensional analogue of the Korteweg-deVries equation. Here we present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions. As an application we show that general classes of equations, which include the Kadomtsev-Petviashvili and the Davey-Stewartson equations, possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct Poisson brackets. Furthermore, we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems.