Showing papers on "Ladder operator published in 1981"
78 citations
TL;DR: In this article, the ergodic and KMS channels are studied by operator algebraic methods and the dynamical properties of these channels are discussed. But the quantum ergodics are not considered.
71 citations
TL;DR: In this article, an asymptotic solution of the Cauchy problem and self-similar solutions for unitary-nonlinear operator equations are constructed for the Hartree equation with Coulomb interaction.
Abstract: Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.
42 citations
25 citations
22 citations
17 citations
TL;DR: In this article, the euclidean Green functions of the 't Hooft vortex operator are studied in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral.
9 citations
9 citations
TL;DR: In this article, a singularity of the differential operator is identified: in it the higher time derivative appears in the same term as the higher derivatives with respect to coordinates, which is called the higher-time derivative singularity.
9 citations
TL;DR: The relationship between the algebra of finite-dimensional matrices acting as raising and lowering operators on a given set of vectors and the operators of multiplication by a variable ξ and of differentiation with respect to ξ are tersely reviewed in this paper.
Abstract: The relationship between the algebra of finite-dimensional matrices acting as raising and lowering operators on a given set of vectors and the operators of multiplication by a variable ξ and of differentiation with respect to ξ are tersely reviewed. A number of matrix formulae implied by this relationship is reported and the possibility to implement them in explicit form is highlighted by referring to a remarkably neat construction of matrices of ordern, defined explicitly in terms ofn arbitrary numbers and satisfying the algebra of raising and lowering operators.
6 citations
TL;DR: The connection between the projection operator approach and the continued fraction approach to perturbation theory is investigated in this paper, where a concise solution to the linear operator equation A mod x>= mod b> is found in terms of the level shift operator using projection operator techniques.
Abstract: The connection between the projection operator approach and the continued fraction approach to perturbation theory is investigated. A concise solution to the linear operator equation A mod x>= mod b> is found in terms of the level shift operator using projection operator techniques. The analogous process to the use of projection operators in the continued fraction method of solving the same problem is identified, and a parallel development performed. The connection between the two approaches is thereby established, and continued fraction expressions for the level shift operator obtained. The abstract equation is then specialised to deal with (i) the eigenvalue problem and (ii) the calculation of transition probabilities for quantum mechanical systems described by a time-independent Hamiltonian. Particular attention is paid to the problem of degeneracy and it is shown that the most convenient expressions are found by a hybrid of the two approaches.
TL;DR: In this article, the perturbed ladder operator method is used to obtain perturbed eigenfunctions in terms of the unperturbed functions of the ladder operator for any type of factorisation (types A to F).
Abstract: The finite difference aspect of the perturbed ladder operator method is reinvestigated. By the use of finite difference calculus, resolution of the factorisability condition is achieved, at any order of perturbation, without assuming for the ladder and factorisation functions any particular dependence on the quantum number. A novel procedure of obtaining perturbed eigenfunctions in terms of the unperturbed functions is described. The method, which holds for any type of factorisation (types A to F), is applied to resolution of the type A wave equation. The perturbed type A problem, which has not been previously treated, contains, as particular cases, the perturbation of the spherical harmonics Ylm (or generalised Yl, gamma m) functions of the symmetric top functions Dmki and, more generally, of the hypergeometric functions F( alpha , beta ; gamma ;x).
TL;DR: In this paper, a local meson operator was constructed from the path-ordered operator on QCD 2, which satisfies the Klein-Gordon equation with the mass identical to the eigenvalue of the 't Hooft equation.
TL;DR: In this paper, a new set of coherent angular momentum states for the oscillator was proposed by diagonalizing v and λ. They are shown to be minimum-uncertainty states for variables v, v + λ, and ǫ+ and provide a new quasiclassical description of the oscillators.
Abstract: The Hamiltonian for the oscillator has earlier been written in the form H=ℏω(2v+v+λ+·λ+3/2), where v+ and v are raising and lowering operators for v+v, which has eigenvalues k (the "radial" quantum number), and λ+ and λ are raising and lowering 3-vector operators for λ+·λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now denned by diagonalizing v and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N 3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables v, v +λ, and λ+ and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.
TL;DR: In this paper, the general form of U(n) raising and lowering operators was obtained and the Hermiticity properties of these operators were investigated, with trivial modifications to the orthogonal groups.
Abstract: It is the aim of this paper to obtain the general form of U(n) raising and lowering operators. The raising and lowering operators constructed previously by several authors are then compared. The Hermiticity properties of these operators are also investigated. The methods presented extend, with trivial modifications, to the orthogonal groups.
TL;DR: In this article, the kinetic theory for a quantum gas with Boltzmann statistics is analyzed for the case when bound pairs occur, in which the processes of molecular formation and breakup, rearrangement collisions, and elastic and inelastic atom-molecule scattering each contribute a nonnegative reaction rate.
Abstract: The kinetic theory for a quantum gas with Boltzmann statistics is analyzed for the case when bound pairs occur. The method used is the binary-collision expansion, applied to the triple-collision operator which occurs in the density-expansion of a Green-Kubo formula. The bound-state contributions are extracted with the aid of the Faddeev analysis of the three-body problem. The results take the form of a binary atom-molecule collision operator, in which the processes of molecular formation and breakup, rearrangement collisions, and elastic and inelastic atom-molecule scattering each contribute a non-negative reaction rate. Reducible diagrams contribute the leading part to rearrangement collisions, and also a correlation correction to the Boltzmann collision operator. The fluxes in the Green-Kubo formula are assumed to be sums of single-particle functions; the atom-molecule collision operator then acts on fluxes which are a sum of an atom term plus a two-particle term obtained by averaging over the molecular state.
TL;DR: The invariant operator range lattices of a wide class of uniformly closed algebras (including C *-algebra) are stable under weak closures as mentioned in this paper, and they are invariant to weak closures.
Abstract: The invariant operator range lattices of a wide class of uniformly closed algebras (including C *-algebras) are stable under weak closures. There is an algebra whose invariant operator range lattice contains properly the corresponding lattice of its norm closure. An operator range transitive algebra is operator range n -transitive for all n. A normal operator is algebraic if and only if each of its invariant operator ranges is the range of some operator commuting with it.
01 Jan 1981
TL;DR: In this paper, the problem of solving linear operator equations with a singular linear operator is ill-posed, if one includes the operator in the data, and sufficient conditions on the perturbations in the operator and the right-hand side that guarantee stability of solutions are given.
Abstract: In general, the problem of solving linear operator equations with a singular linear operator is ill-posed, if one includes the operator in the data. We give sufficient conditions on the perturbations in the operator and the right-hand side that guarantee stability of solutions. With different techniques we treat the cases that the linear operator is a Fredholm operator with index zero and that the linear operator is compact.
TL;DR: The algebraic regularity of monopole eigenfunctions in various dynamical O (2,1) bases shows the symmetry in radial phase space as discussed by the authors, and the multipole tensors serve as ladder operators between different O (1)⊗ O (3) quasi-bound states.