Table of contents

A method based on Fredholm approximants is proposed for the approximate solution of the Lippmann-Schwinger equation. All diagonal Fredholm approximants constructed in this way are shown to be unitary. The method can be generalized to include Bethe-Salpeter and N/D equations.

The susceptibility of the Ising S=1 model with biquadratic interactions is expanded in a high temperature series on the FCC lattice for a range of the biquadratic interaction parameter that includes the Potts model (1952). A tricritical point and a point above which the magnetization M remains zero are approximately located. The Potts model seems to fall in the region of the first-order phase transition in M.

A general method for computing critical points of planar two-dimensional Ising models is outlined. As examples the critical points of a pair of archimedean lattices are computed.

Investigating by series analysis the polarization of the eight-vertex model as a function of electric field at T_c and zero magnetic field, estimates of the exponent delta _e are obtained. These estimates are combined with a conjectured scaling relation to give a simple prediction for the spontaneous polarization exponent beta _e.

The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)=¹/₂p_jg^jk(q)p_k+¹/₂(A^j(q),p_j)+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.

A simple unified procedure, based on a generalization of a method first used by Weinstein in 1934, yields effective lower and upper bounds to energy levels and overlap integrals of quantum-mechanical systems. A further generalization yields bounds to expectation values.

A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H₀(t)=H₀+H₁(t) and the full hamiltonian H(t)=H₀(t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega _+or-(s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H₁(t) of the type H₁(t)=H₁ cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H₁(t) is absent.

For pt. I see ibid., vol.7, no.5, 572 (1974). The general theory of the scattering of two particles in a time-dependent external field, presented in paper I, is applied to the case of two point particles with internal structure which interact through a potential with matrix representation (V_kl(r)). Under suitable, quite general, conditions upon the l² operator bound mod V(r) mod of (V_kl(r)) the existence of the wave operators Omega _+or-(s) is proven for two field configurations. The first is that of a spatially homogeneous time-dependent field, whereas in the second case the field is inhomogeneous but localized in space. Some attention is paid to the problem of the existence of differential cross sections and their relation to the scattering operator S(s)= Omega *₊(s) Omega _-(s).

The prescription given by Gautreau and Hoffman (1970) to generate axially symmetric electrovac solutions from the Weyl vacuum solutions in general relativity, is shown to be completely equivalent to the procedure given earlier by Harrison (1965) yielding, thus, only a known class of solutions of the Einstein-Maxwell equations.

Considerations based on the symmetry inherent in Maxwell's electrodynamics enable problems involving doubly anisotropic media with non-parallel principal axes of epsilon and mu to be tackled. It is shown that one can pass on from the results of single anisotropy to the more general double anisotropy with the aid of a set of substitutions. These are then applied to obtain expressions for the radiation cones and energy loss of a charge moving uniformly in a generalized uniaxial medium.

The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.

The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.

It is shown that a nonlinear integral equation for turbulent energy transport may be reinterpreted in terms of a Heisenberg-type effective viscosity. A new equation is derived for the effective viscosity. This is found to permit general expansions of the integral kernels, in powers of wavenumber ratios, leading to local (differential) equations for the energy spectrum and effective viscosity. It is found that these equations yield the Kolmogoroff distribution as the inertial-range solution, and that the numerical predictions agree quite well with experimental results. The final equations are similar to equations recently derived by Nakano (1972), and the relationship between the two theories is discussed.

The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k_BT) along the coexistence curve for the specific heat C_H, the magnetization M, and its first five derivatives delta ¹M/ delta mu ¹, where mu =exp(-2mH/k_BT). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta ^lM/ delta z^l. Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.