arXiv subjects
Publications and source records attributed to Yu. F. Smirnov.
The generators of the algebra $gl_{n+1}$ in a form of differential operators of the first order acting on ${\bf R}^n$ with matrix coefficients are explicitly written. The algebraic Hamiltonians for a matrix generalization of $3-$body Calogero and Sutherland models are presented.
For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.
We study some q-analogues of the Racah polynomials and some of their applications in the theory of representation of quantum algebras.
The structure positive of unitary irreducible representations of the noncompact $u_q(2,1)$ quantum algebra that are related to a positive discrete series is examined. With the aid of projection operators for the $su_q(2)$ subalgebra, a $q$-analog of the Gel'fand--Graev formulas is derived in the basis corresponding to the reduction $u_q(2,1)\to su_q(2)\times u(1)$. Projection operators for the $su_q(1,1)$ subalgebra are employed to study the same representations for the reduction $u_q(2,1)\to u(1)\times su_q(1,1)$. The matrix elements of the generators of the $u_q(2,1)$ algebra are computed in this new basis. A general analytic expression for an element of the transformation bracket $ _q$ between the bases associated with above two reductions (the elements of this matrix are referred to as $q$-Weyl coefficients) is obtained for a general case where the deformation parameter $q$ is not equal to a root of unity. It is shown explicitly that, apart from a phase, $q$-Weyl coefficients coincide with the $q$-Racah coefficients for the $su_q(2)$ quantum algebra.
The calculation of Franck-Condon factors between different one-dimensional Morse potential eigenstates using a formula derived from the Wigner function is discussed. Our numerical calculations using a very simple program written in Mathematica is compared with other calculations. We show that our results have a similar accuracy as the calculations performed with more sophisticated methods. We discuss the extension of our method to include non-Condon effects in the calculation.
The structure of the reported excitation spectra of the light unflavored baryons is described in terms of multi-spin valued Lorentz group representations of the so called Rarita-Schwinger (RS) type (K/2, K/2)* [(1/ 2,0)+ (0,1/2)] with K=1,3, and 5. We first motivate legitimacy of such pattern as fundamental fields as they emerge in the decomposition of triple fermion constructs into Lorentz representations. We then study the baryon realization of RS fields as composite systems by means of the quark version of the U(4) symmetric diatomic rovibron model. In using the U(4)/ O(4)/ O(3)/ O(2) reduction chain, we are able to reproduce quantum numbers and mass splittings of the above resonance assemblies. We present the essentials of the four dimensional angular momentum algebra and construct electromagnetic tensor operators. The predictive power of the model is illustrated by ratios of reduced probabilities concerning electric de-excitations of various resonances to the nucleon.
We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U_q(su(3)). We obtain a very compact general analytical formula for the U_q(su(3)) CGCs in terms of the U_q(su(2)) Wigner 3nj-symbols.
The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.
A symmetry of the parameter space of interacting boson models IBM-1 and IBM-2 is studied. The symmetry is associated with linear canonical transformations of boson operators, or, equivalently, with the existence of different realizations of the symmetry algebras of the models. The relevance of the parameter symmetry to physical observables is discussed.
We discuss the symmetry of the parameter space of the interacting boson model (IBM). It is shown that for any set of the IBM Hamiltonian parameters (with the only exception of the U(5) dynamical symmetry limit) one can always find another set that generates the equivalent spectrum. We discuss the origin of the symmetry and its relevance for physical applications.
We study in detail the bound state spectrum of the generalized Morse potential~(GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schrödinger equation with the Laplace equation on the hyperboloid and the Schrödinger equation for the Pöschl-Teller potential, we explain the exact solvability of the problem by an $so(2,2)$ symmetry algebra, and obtain an explicit realization of the latter as $su(1,1) \oplus su(1,1)$. We prove that some of the $so(2,2)$ generators connect among themselves wave functions belonging to different GMP's (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, $so(2,2)$ belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.
A purely group-theoretical approach (for which the symmetric group plays a central rôle), based upon the use of properties of fractional-parentage coefficients and isoscalar factors, is developed for the derivation of the Coulomb energy averaged over the states, with a definite spin, arising from an atomic configuration $n \ell^N$.
Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$ω$. By considering the latter together with its partner corresponding to the replacement of~$ω$ by~$-ω$, we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.
A set of compatible formulas for the Clebsch-Gordan coefficients of the quantum algebra $U_{q}({\rm su}_2)$ is given in this paper. These formulas are $q$-deformations of known formulas, as for instance: Wigner, van der Waerden, and Racah formulas. They serve as starting points for deriving various realizations of the unit tensor of $U_{q}({\rm su}_2)$ in terms of $q$-boson operators. The passage from the one-parameter quantum algebra $U_{q }({\rm su}_2)$ to the two-parameter quantum algebra $U_{qp}({\rm u}_2)$ is discussed at the level of Clebsch-Gordan coefficients.
The content of this work is concerned with Jordan-Schwinger calculus using $q$-deformed bosons.
We report in this article three- and four-term recursion relations for Clebsch-Gordan coefficients of the quantum algebras $U_q(su_2)$ and $U_q(su_{1,1})$. These relations were obtained by exploiting the complementarity of three quantum algebras in a $q$-deformation of $sp(8, \gr)$.
Properties of the neutron rich $^{11}$Li nucleus are calculated in the framework of the cluster model $^{9}$Li $+n+n$. The formalism of the harmonic oscillator representation of the scattering theory is used for the description of bound and continuum spectrum states in the three-body-democratic-decay approximation. It is shown that this approach allows one to take into account adequately the long asymptotic tail of the $^{11}$Li wave function ({\em neutron halo}) and to reproduce correctly the binding energy, radius and $^{11}$Li electromagnetic dissociation cross-section on target nuclei. The shape and the energy position of the $B(E1)$ peak corresponding to the soft dipole mode are also in agreement with experiment.
We show that a quantum system with nonlocal interaction can have bound states of unusual type (isolated states (IS)). IS is a bound state that do not generate a $S$-matrix pole. IS can have positive as well as negative energy and can be treated as a generalization of bound states embedded in continuum on the case of discrete spectrum states. Formation of IS in the spectrum of quantum system is studied using a simple rank--2 separable potential with harmonic oscillator formfactors. Some physical applications are discussed, in particular, we propose separable $NN$ potential that describes not only most important two-nucleon data (deuteron binding energy and s-wave triplet and singlet scattering phases) but also the trinucleon binding energy without making use of three-body forces.