Showing papers on "Ladder operator published in 1969"

On Operator Algebras and Invariant Subspaces

Abstract: If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies . Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.) ARVESON's THEOREM. If is a weakly closed algebra which contains an m.a.s.a.y and if Lat , then is the algebra of all operators on . A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

Quantum-mechanical conjugate of the hamiltonian operator

Abstract: Conjugate functions of various classical Hamiltonians are studied and different methods for their construction are presented. From these classical functions, for many dynamical systems, Hermitian operators can be found which depend linearly on time and are conjugates of Hamiltonian operators. The eigenvalues of conjugate operators, under certain conditions, represent the result of measurement of time in nonrelativistic quantum mechanics. Because of this property « time operators » may be used to study the quantum theory of microscopic clocks and to determine their accuracy. For noninteracting relativistic particles with spin zero and spin one-half a direct method of obtaining time operator is given. These conjugate operators have been used to derive well-known relations of the time delay and the rate of change of phase shift with energy in scattering theory.

On one parameter families of real solutions of nonlinear operator equations

Abstract: 10. , A necessary condition that a cellular upper semicontinuous decomposiion ofE yield E, Trans. Amer. Math. Soc. 122 (1966), 427-435. 11. , Decompositions of S and pseudo-isotopies, Notices Amer. Math. Soc. 15 (1968), 103. 12. W. Voxman, Decompositions of 3-manifolds and pseudo-isotopies, Notices Amer. Math. Soc. 15 (1968), 547. 13. , On the shrinkability of decompositions of 3-manifolds, Notices Amer. Math. Soc. 15 (1968), 649.

Is there an Azimuthal Angle Observable

Abstract: The concept of an azimuthal-angle operator canonically conjugate to the z component of angular-momentum operator is known to present certain difficulties in quantum mechanics. These difficulties are resolved rigorously for the first time, and their implications are discussed.

On Continuous Eigenvalues in Neutron Thermalization

Abstract: The continuous eigenvalue spectra of the linearized Boltzmann operator describing the energy distribution of neutrons in an infinite Einstein crystal are studied This operator consists of two terms: a multiplication operator and an integral operator with δ‐function‐type singular kernel The eigenvalue problem is transformed into the solution of an inhomogeneous integral equation by applying Case's method on the one‐velocity transport equation The existence of the solution of the integral equation is examined by the Neumann‐series expansion It is found that for sufficiently low temperature the range of numerical values of the multiplication operator forms the continuous eigenvalue spectra of the Boltzmann operator and the corresponding eigenfunctions are of δ‐function type

Uniqueness of the Hamiltonian in quantum field theories

Abstract: In most quantum field theories, one defines the Hamiltonian (energy) operatorH as a limit of “cutoff” operators $$H_s :H = \mathop {\lim }\limits_{s \to \infty } H_s $$ . (The operatorH s would be the correct Hamiltonian for a world in which all momenta are smaller thans.) Since the cutoff operators seldom converge in any of the standard operator topologies, it is often necessary to invent more subtle notions of “convergence”. For some of the these, it is not obvious that the “limit” operatorH is unique. In this note we point out that for one such method of obtaining convergence, the “limit” operator isnot unique. In fact, (under mild assumptions about the operatorsH s ), ifH s converges toH, thenH s also converges toH+R, whereR is an arbitrary bounded positive operator.

Field operator for an interacting unstable elementary system

Abstract: The field operator ϕj;α (x;s), describing an interacting unstable elementary system is introduced. Asymptotic fields are represented by free fields with continuous additional parameters. Consistency with the Haag-Ruelle theory of asymptotic states is achieved.