Abstract: The Fock space Hamiltonian H has a simpler structure than its projection Hn to n‐particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: ...

Abstract: A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.

Abstract: In this paper we prove existence for all time t≦0, uniqueness, regularity and stability of solutions of a Cauchy problem in a Hilbert space H for the equation when δ is a positive constant sufficiently large, A an operator in H and M a given function, satisfying convenient hypothesis. An application is given; in particular, existence for all t≦0, uniqueness and stability of classical solutions of an initial-boundary value problem for the nonlinear partial integro-differential equation (βk positive constants) is obtained. AMOS(MOS) No. 35L20; 35B40

Abstract: An account of the quantum theory of the molecular hypothesis for chemical substances is presented, and is used as a basis for a critical discussion of the theory of natural optical activity. Atoms and molecules are characterized as composite elementary excitations (or quasi-particles) of the macroscopic quantum-mechanical system we call matter, and the spontaneously broken space inversion symmetry revealed by the existence of optical isomerism is studied in this context. Special attention is paid to the representation of the space-inversion operator {ie1-1}. Finally a variety of microscopic quantum theories of natural optical activity are critically reviewed.

Abstract: The basic relations of the Weyl-Wigner representation of quantum mechanics are reviewed. It is stressed that this representation is unique and based on a phase-space operator which corresponds to an observable in Dirac's sense.

Abstract: CONTENTSIntroduction § 1. The finite-dimensional case § 2. Stochastic semigroups in the L2-strong theory § 3. Homogeneous strongly continuous semigroups with the group of the first moments § 4. Stochastic equations of diffusion type with constant coefficients § 5. Continuous homogeneous stochastic semigroups in the presence of two moments References

Abstract: A general expression for the electron-scattering coincidence cross section for the reaction A1(e, e′ X) A2 with a nuclear target is derived in the one-photon exchange approximation. The result is exact to lowest order in α, the fine-structure constant. It is expressed in terms of four kinematic factors involving the electron scattering variables in the laboratory frame, and four combinations of transition matrix elements of the nuclear current operator expressed in the center-of-momentum (COM) frame. The nuclear matrix elements are decomposed into transition amplitudes of definite angular momentum using a helicity analysis. General expressions for the angular distribution of particle X in the COM frame are then derived. The analysis is independent of the detailed structure of the nucleus and particle X and depends only on general symmetry considerations and the existence of a local electromagnetic current operator for the hadronic target. A unitary transformation from the helicity basis for the final particle X and A2 to an LS coupling basis is relevant if X is massive and a finite number of total angular momenta J are involved in the reaction. Tables of angular correlation coefficients are given for the case where the initial nucleus A1 has J1π = 0+. They constitute one of the most useful results of this paper. Connection is made in the “static limit,” and with the assumption that the reaction proceeds through a finite number of Breit-Wigner resonances with a corresponding factorization of the electroproduction transition matrix elements, to the familiar electromagnetic transition multipole moments involving excitation of a nuclear state Jπ. The relation to previous work by de Forest and by Drechsel and Uberall is discussed. Analytic expressions for the coincidence cross sections are given for spin-zero systems and some very simple, basic models of nuclear “giant resonance” excitations. It is hoped that they will be useful in obtaining insight into the coincidence cross section and in planning future experiments. Finally, a reanalysis of the recent Stanford data of Calarco et al. on 12 C (e, e′ p 0 ) 11 B ( 3 2 − ) in the vicinity of the giant dipole resonance in 12C is carried out using a very simple nuclear model but retaining all the terms in the coincidence cross section, some of which were previously neglected.

Abstract: A quasistatic theory of the dipole autocorrelation function for isolated molecular spectral lines has been developed and applied to linear molecules perturbed by atoms and molecules The matrix elements of the evolution operator associated with the molecular dipole moment are treated to all orders in the interaction without any cutoff procedure An analytical expression of the intensity distribution in terms of the intermolecular potential has been obtained which is easily tractable for involved molecular systems Numerical tests of this theory have been performed for rotation–vibration lines of HCl–Ar, HCl–Xe, and CO2–Ar gas mixtures by comparing the present results with that of a unified theory of the line shape for molecule–atom couples, valid from resonances to the far wings and including all orders in the interaction These tests justify the present approach and show a considerable reduction of the computation time required (by more than two orders of magnitude), allowing now the study of the polyatomic preturbers case Application to infrared absorption lines of HCl self‐perturbed and perturbed by CO2 leads to a good agreement with the experimentally determined super‐Lorentzian profile

Abstract: We characterize the class of linear operators on a finite dimensional inner product space which are the exponents of a full operator-stable law. This answers a question of Paulauskas [6] concerning those operator-stable laws whose characteristic functions are the exponential of quadratic forms. The symmetry group of such laws must be conjugate to the group of all orthogonal transformations on the space.

Abstract: A careful analysis of the assumptions and approximations underlying the derivation of the usual kinetic equation for a Rayleigh gas (or a Brownian particle) is performed The passage from the exact Boltzmann collision operator to its approximate differential form is thus investigated It is shown that the exact operator can be replaced by the approximate differential one only when proper conditions on the initial heavy-particle velocity distribution are satisfied From this analysis it follows that the usual kinetic equation is unable to describe the initial stages of the relaxation of an initial δ-function or of a maxwellian distribution at a temperature much lower than — or, also, for non-maxwellian interactions, much higher than — the equilibrium temperature In any case, the ratio of the initial velocity distribution to the equilibrium one cannot present a fine structure, or too appreciable deviations from a polynomial form of first or second degree in the velocity components, in velocity-space regions which have linear dimensions which are not large compared with the heavy-particle root-mean-square velocity change per elastic collision Moreover, except that in the Maxwell model, the initial mean energy of the heavy particles cannot too largely exceed the equilibrium value

Abstract: In this chapter, we are studying operator equations having a linear part in a different way. This allows us to use the classical method of iteration which is different from the method of majorant series. This study allows us to give new results and new simple proofs of some known results such that the classical Poincare theorem and also a more clear presentation of a theorem due to Kaplan. Finally, we are giving a proof of the classical Siegel’s theorem (case of small denominators).

Abstract: After a preliminary functional study of the operator associated with the relevant Boltzmann equation, which is shown to be a contraction operator, a nonlinear integral evolution problem occurring in the diffusion of the particles of a mixture is solved by resorting to a rigorous iterative scheme, in the case without removal. According to this scheme, an explicit recursive representation for the general iterated solution of order n is developed. Structure and behavior of the solution so obtained are investigated and commented on.

Abstract: A kinetic equation in operator form for the one body density matrix is derived by means of a convenient truncation of the quantal Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Using projection techniques in a procedure similar to the derivation of the master equation, the evolution of the diagonal and the nondiagonal terms of the density matrix can be disentangled. A quantal, non-Hermitian master equation for the diagonal part is extracted and it is seen that corrections to the collision frequency or to the relaxation time arise as a consequence of the creation and destruction of incoherent events.

Abstract: Let M be a compact Riemannian manifold equipped with Laplace–Beltrami operator Δ. We use the Rademacher-Menchoff theorem and the asymptotics of the eigenvalues of Δ to show that if a function belongs to an L 2 Sobolev space of positive index then its expansion, in terms of eigenfunctions of Δ, converges almost everywhere on M .

Abstract: I The Rediscovery of a Mathematical Tool.- I.1 Max Born's Interpretation of Heisenberg's Quantum Condition.- I.2 The Development of Matrix Calculus.- I.3 Early Applications of Matrix Methods in Physics.- I.4 Born's New Collaborator: Pascual Jordan.- II Matching the Tools and the Task.- II.1 The Programme of Matrix Mechanics.- II.2 Operations with Matrices.- II.3 Dynamical Laws and Energy Conservation.- II.4 An Example of Discrete Mechanics: The Oscillator.- II.5 Preliminary Remarks on Radiation.- III Completion of the Matrix Scheme.- III.1 The Three-Man Collaboration.- III.2 Towards a New Perturbation Theory.- III.3 Several Degrees of Freedom and Degeneracy.- III.4 Born's Idee Fixe and a Letter to Niels Bohr.- III.5 The Eigenvalue Problem and the Transformation to Principal Axes.- III.6 Continuous Spectra and the Significance of the Transformation Matrix.- IV The Success of Matrix Mechanics.- IV.1 The Treatment of Dispersion Phenomena.- IV.2 Fluctuations in Cavity Radiation.- IV.3 The Conservation of Angular Momentum.- IV.4 Wolfgang Pauli's Conversion.- IV.5 The Solution of the Hydrogen Problem.- IV.6 The Problems of Intensities and the Diatomic Molecule.- V Modifications and Extensions of Matrix Mechanics.- V.1 Nonmechanical Stress versus Spin.- V.2 Field-Like Representation of Quantum Mechanics.- V.3 The Operator Mechanics.- V.4 Multiply Periodic Systems: Action-Angle Variables and the Method of Complex Integration.- V.5 The Electron Spin, Fine Structure and Anomalous Zeeman Effects.- V.6 Key to the Helium Problem.- References.- Author Index.

Abstract: An effective method for solving the three-body problem with Coulomb interaction is presented systematically. The essential feature of the method is an expansion of the wave function of the three-particle system with respect to an adiabatic basis and reduction of the original Schroedinger equation to a system of ordinary differential equations. Convergence of the adiabatic expansion is ensured not only by the smallness of the ratio of the particle masses but also by the smallness of the nondiagonal matrix elements of the kinetic-energy operator of particles of the same charge. The possibilities of the method are demonstrated by the example of the calculation of the energies and wave functions of all states of the ..mu..-mesic molecules of the hydrogen isotopes and the e/sup -/e/sup -/e/sup +/ system. The method is equally suitable for calculating the ground state and the excited states of a three-particle system. This is particularly important in the calculation of the energies of the weakly bound states of the mesic molecules dd..mu.. and dt..mu.., knowledge of which is needed to describe the processes of muonic catalysis of nuclear fusion reactions.

Abstract: The statistical operator of von Neumann and its use by Eberly and Singh to give a natural formulation of energy–time uncertainty are explained in elementary detail.

Abstract: This paper derives a number of sum rules for nonrelativistic three-body scattering. These rules are valid for any finite region ..sigma.. in the six-dimensional coordinate space. They relate energy moments of the trace of the onshell time-delay operator to the energy-weighted probability for finding the three-body bound-state wave functions in the region ..sigma... If ..sigma.. is all of the six-dimensional space, the global form of the sum rules is obtained. In this form the rules constitute higher-order Levinson's theorems for the three-body problem. Finally, the sum rules are extended to allow the energy momtns have complex powers.

Abstract: In this letter we show that the linear equations for the gravitational field with Heavisidian monopoles could be expressed in quaternion form using a quaternion conjugate operator as well as quaternion differential operator.