Pseudospin and nuclear deformation

An operator which effects the transformation to the pseudospin representation within the oscillator shell model at arbitrary triaxial deformation is constructed as a special projection of the momentum helicity operator. Since the exact transformation of many important operators cannot be performed in a closed analytical form, a procedure of approximate transformation is developed. In the spherical and asymptotic prolate limits the transforms thus derived reduce to the known exact results. The approximate transform of a modified Nilsson Hamiltonian is found to be almost indistinguishable from the “pseudo” Hamiltonian with the strongly reduced spin-orbit strength.     

The index of the scattering operator on the positive spectral subspace

We construct the scattering operator for a spinor field in a time dependent background by the Dyson expansion. Then we show that the restriction of the scattering operator to the positive spectral subspace (with respect to a reference Hamiltonian) is Fredholm. The computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero. We obtain the index in terms of a cohomological formula by means of the Atiyah-Singer index theorem.     

Weyl transformation in Fermi systems

In the path integral representation, the Hamiltonian in a quantum system is associated with the Hamiltonian in a classical system through the Weyl transformation. From this, it is possible to describe the time evolution in a quantum system by the Hamiltonian in a classical system. In a Bose system, the Weyl transformation is defined by the eigenstates of the canonical operators, since the Hamiltonian is given by a function of the canonical operators. On the other hand, in a Fermi system, the Hamiltonian is usually described by a function of the creation and annihilation operators, and hence the Weyl transformation is defined by the coherent states which are the eigenstate of an annihilation operator. Here, we formulate the Weyl transformation in Fermi systems in terms of the eigenstates of the canonical operators so as to clarify the correspondence between both systems. Using this, we can derive the path integral representation in Fermi systems.     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Hamiltonian operator with three arbitrary constants is presented. When the arbitrary constants in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulations a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

Unitary transformation treatment of a single hole in an Ising antiferromagnet

The behavior of a single hole in a two-dimensional Ising antiferromagnet (t-J ^z model), is studied in the generalized Dyson-Maleev representation, where the spins are mapped on boson operators and the hole is described as a spinless fermion. The formal similarity with Fröhlich's polaron Hamiltonian suggests that thet-J ^z model can be approximately diagonalized by means of two successive unitary transformations, analogous to those used by Lee, Low, and Pines in their intermediate-coupling treatment of the polaron. Our approach yields an upper bound to the exact ground state energy, as well as the corresponding ground state eigenvector. Fork=0 our energy bound is remarkably close to the result of the self-consistent Born approximation over a wide range of the coupling parameter, which includes the range typically assumed for the high-T _c materials. The ground state eigenvector is used to calculate the spatial distribution of bosons (spin deviations) surrounding the hole. Here our results are qualitatively very similar to those obtained in previous work, showing that our ground state eigenvector accounts quite well for the small size of the “spin polaron” in thet-J ^z model.     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Ha-miltonian operator with three arbitrary constants is presented. When the arbitrary constants -in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulation, a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

双模积分型投影算符及其应用

引进了幺正的双模积分型投影算符,利用有序算符内的积分(IWOP)技术分析了其变换特性;然后利用该积分型投影算符对角化了双模耦合量子谐振子体系的哈密顿量,从而求出了体系的本征能级与本征波函数;最后讨论了特例情形.     

双模坐标-动量积分型投影算符及其在量子光学中的应用

引进了幺正的双模坐标-动量积分型投影算符,利用有序算符内的积分（IWOP）技术分析了其变换特性,并导出了其正规乘积展开式.然后利用该积分型投影算符对角化了双模耦合量子谐振子体系的哈密顿量,从而求出了体系的本征能级与本征波函数.最后讨论了特例情形. 关键词： 积分型投影算符 有序算符内的积分技术 坐标-动量耦合     

Deductive inference of nuclear boson models (IBM, TQM, FQM) based on SU(6) dynamical symmetry

A single deductive inference of Schwinger realization (= interacting boson model—IBM), Holstein-Primakoff realization (= truncated quadrupole phonon model—TQM) and Dyson realization (= finite quadrupole phonon model—FQM) of dynamical SU(6) quadrupole collective algebra (QCA) is presented with a full scope of their isomorphism on the level of representations. Dyson realization of QCA is explicitly constructed by using holomorphically parametrized generalized coherent state and explicit form of root vectors. Utilizing appropriate orthogonalizing operators Holstein-Primakoff realization of QCA has been derived from the Dyson realization. The carrier spaces of Schwinger and Holstein-Primakoff realizations are investigated on the same footing and Marshalek's boson is rigorously derived. The intertwining operator which connects Schwinger and Holstein-Primakoff realizations is constructed and its domain and image are determined. It is shown that the intertwining operator has well-defined inverse in a definite factor space of the IBM basis space which is proved to be isomorphic to the physical subspace of the TQM basis space, meaning equivalence of IBM and TQM on level of representations.     

Multiple-quantum operator algebra spaces and description for unitary time evolution of multilevel spin systems

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.     

Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.     

Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group

We find that the coherent state projection operator representation of symplectic transformation constitutes a loyal group representation of symplectic group. The result of successively applying squeezing operators on number state can be easily derived.     

Spectral decompositions of operators on non-Archimedean orthomodular spaces

The most central property of an infinite-dimensional Hilbert space is expressed by the projection theorem: Every orthogonally closed linear subspace is an orthogonal summand. Besides the obvious Hilbert spaces, there exist other infinite-dimensional orthomodular spaces. Here we study bounded linear operators on an orthomodular spaceE constructed over a field of generalized power series with real coefficients. Our main result states that every bounded, self-adjoint operator gives rise to a representation ofE as the closure of an infinite orthogonal sum of invariant subspaces each of which is of dimension 1 or 2. The proof combines the technique of reduction modulo the residual spaces with theorems on orthogonal decompositions of finite matrices over fields of power series.     

Coherent State Projection Operator Representation of Symplectic Transformations as a Loyal Representation of Symplectic Group

We find that the coherent state projection operator representation of symplectic transformation constitutesa loyal group representation of symplectic group. The result of successively applying squeezing operators on numberstate can be easily derived.     

量子棒中强耦合杂质束缚极化子的振动频率

给出了具有椭球边界量子棒经坐标变换成球形边界的哈密顿量。采用线性组合算符和幺正变换的方法研究了在非均匀抛物限制势下量子棒中强耦合杂质束缚极化子的性质。导出了量子棒中强耦合杂质束缚极化子的振动频率和声子平均数随库仑束缚势、电子-声子耦合强度、椭球的纵横比和量子棒的横向和纵向有效受限长度的变化关系。数值计算结果表明:振动频率和声子平均数随电子-声子耦合强度和库仑束缚势的增强而增加,随量子棒的横向和纵向有效受限长度和椭球的纵横比的减小而增大。表现出量子棒的奇特的量子尺寸限制效应。     

The temperature dependence of rotational tunnelling

A novel approach to the temperature dependence of rotational tunnelling is presented. By means of a projection operator technique a generalized Master equation is derived, where the effects of coupling to lattice vibrations are embraced in a memory function non-local in time. After an expansion in a power series in the coupling Hamiltonian, the Master equation is used for the calculation of two-times correlation functions of scattering operators, the Fourier transforms of which give the scattering function. The theory allows for all features of the spectrum obtained by neutron scattering methods, in particular for those of the central peak and the librational excitations. Furthermore, it is not confined to the low temperature regime, but rather covers the whole range of temperatures of experimental interest.     

Quantum Brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.     

Quantum dynamics of a generalized Coleman-Hepp model

Studies on a generalized Coleman-Hepp model are done on the basis of a spin coherent state representation and a transformation property of the model Hamiltonian. Namely, transforming the original model Hamiltonian into a simpler form, we can determine time evolution of the whole system by successive applications of rotation operators in a spinor space. Dynamics of detector spins as well as that of an incident particle are fully discussed. Explicit numerical evaluations are also performed. Relevance of our solution to a generalized Cini model is also briefly mentioned. Received 24 August 1999     

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ≤ft\vert z\right\rangle _{g} and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens--Fresnel integration transformation describing optical diffraction). The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz-rz^{\ast } in the % ≤ft\vert z\right\rangle _{g} representation, while ≤ft\vert z\right\rangle _{g} in phase space is graphically denoted by an ellipse.