Effect of electron correlation on the forced electric dipole transition probabilities in fn systems

A method for the investigation of the effects of electron correlation on the forced electric dipole transition probabilities between levels of the fⁿ configuration is presented. The approach is based on double perturbation theory and on the application of effective operators. Effective operators describing the correlation effects are generated for various classes of perturbing configurations in terms of generalized unit tensor operators. A classification scheme for the possible operators is presented. In general, one- and two- particle operators are obtained. The latter do not appear in the standard Judd-Ofelt theory. One-particle operators dependent on unit tensor operators of odd rank are also new. The other one-particle operators are of the form considered in the Judd-Ofelt theory. However, the present results indicate that the interpretation of the empirical parameters of this theory should be modified to take into account electron correlation effects.     

Comments on the properties of one- and two-body double tensor operators

Recently, the parity, charge, time and Hermitian conjugation properties of one-body tensor operators have been reexamined and the Racah algebra has been formally extended to two-body tensor operators. In this paper we consider several important aspects related to these problems. First, we present a proof of the communication relations for two-body operators which was previously postulated. Then, comparing the reduced matrix elements for one- and two-body operators we find the normalization constant for the latter. We subsequently show that the P- and T-conjugation relations for tensor operators are relativistically invariant. Finally, we analyze the question of Hermitian conjugation of double tensor operators and conclude that previously stated conditions on their ranks were too restrictive and resulted in omissions of some terms from theoretical analyses. We show examples of tensor operators which were customarily neglected but indeed should be retained in the effective Hamiltonian.     

Algebraic properties of the Dirac equation

The first-order symmetry operators of the Dirac equation are classified according to their tensor properties under transformations of the homogeneous Lorentz group; a minimal system of generators for the ring of symmetry operators of the free Dirac equation is obtained, and the physical meaning of the spin operators is considered; fields are found which admit symmetry operators of first order.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 84–89, February, 1972.The author is grateful to V. N. Shapovalov for discussions and valuable suggestions.     

Coherent States for the Asymmetric Penning Trap

After identifying the appropriate ladder operators, the coherent states for the asymmetric Penning trap are derived as common eigenstates of the annihilation operators. They will be compared with the ones obtained by acting the displacement operator onto the extremal state, as well as with those which minimize some Heisenberg uncertainty relationships. The time evolution and relevant mean values for some operators in these states will be evaluated.     

A note on possibility of existence of the stationary state scattering theory for the density operators

The aim of this paper is an attempt to construct the scattering theory in terms of density operators. The wave operators in the space of trace class linear operators are defined. It is shown that the stationary state theory may be formulated in this space for the H-bounded perturbations.     

Evolution of entanglement in the Heisenberg XY model: the Yangian algebra method

We construct transition operators in terms of generators for Yangian and act with them on the entangled state of the Heisenberg XY model. Different parameter values of the transition operators lead to different evolutions of the entanglement degree of the final states. It is interesting and worth noting that for some special parameters in the transition operators, our calculation is in good agreement with the experiment. This result sheds a new light on the physics of Y(sl(2)) in quantum information.     

关于构造阶梯算符

给出有关构造阶梯算符的定理,提供了构造阶梯算符的一种规范方法。     

Intrinsic normal-ordered vertex operators from the multiloop N-tachyon amplitude

We construct vertex operators for arbitrary mass level states of the closed bosonic string. Starting from a generalization of the Koba-Nielsen amplitude which is suitable for an arbitrary genus Riemann surface, we read the vertex operators from the residues of the poles for the intermediate states. Since the original expression is metric independent and normal ordered without the need of inventing any regularization scheme, our vertex operators also possess these properties. We discuss their general features.     

General chiral gauge theories on the lattice

We still extend the large class of Dirac operators decribing massless fermions on the lattice found recently, only requiring that such operators decompose into Weyl operators. After deriving general relations and constructions of operators, we study the basis representations of the chiral projections. We then investigate correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and their consequences, and getting the precise behaviors under gauge transformations and CP transformations. Various further developments include considerations of the explicit form of the effective action and of a representation of the general correlation functions in terms of alternating multilinear forms. For comparison we also consider gauge-field variations and their respective applications. Finally we compare with continuum perturbation theory.     

Baxter Q-operators of the XXZ chain and R-matrix factorization

We construct Baxter operators as generalized transfer matrices being traces of products of generic R matrices. The latter are shown to factorize into simpler operators allowing for explicit expressions in terms of functions of a Weyl pair of basic operators. These explicit expressions are the basis for explicit expression for Baxter Q-operators and for investigating their properties.     

On the SO(7) symmetry of the SO(8) monopole and quadrupole pairing model

The SO(7) D-pairing symmetry of the SO(8) monopole and quadrupole pairing model is shown to be unusual in that there is no corresponding group generated only by number-conserving operators. Rather, the SO(7) symmetry is manifested in terms of number-conserving operators only by taking a particular combination of the Casimir operators appropriate for the other subgroup chains.     

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

Traces of Intertwining Operators for the Yangian Double

The traces over infinite-dimensional representations of the central extended Yangian double for the product of operators which intertwine these representations are calculated. For the special combinations of the intertwining operators, the traces are identified with form factors of local operators in the SU(2)-invariant Thirring model. This identification is based on the identities which are deformed analogs of the Gauss–Manin connection identities for the hyperelliptic curves.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential

We prove the complete exponential localization of eigenfunctions for the 1-D discrete Schrödinger operators with quasi-periodic potentials having two basic frequencies. It is shown also that for such operators there is no forbidden zones in the spectrum, unlike the operators with one basic frequency.Dedicated to Roland Dobrushin     

Chiral Asymmetry and the Spectral Action

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.     

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.     

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lamé operators.     

Scattering for the wave equation on the Schwarzschild Metric

We study the wave equation for the Schwarzschild metric. Wave operators are constructed which yield solutions with given asymptotic behavior either at infinity or on the horizon. We prove asymptotic completeness for these wave operators.Supported by NSF grant No. PHY82-204399.     

Step Operators for the Quasi-exactly Solvable Systems

In this paper, we investigate the step operators for the quasi-exactly solvable problems. We also discuss the commutation relations between the step operators and the Hamiltonian of the quasi-exactly solvable system. After obtaining the general results, we take the anharmonic oscillators with x ⁶ anharmonicity in quasi-exactly solvable problems as examples to give the specific forms of step operators.