Volume 15, Issue 2, February 1974

The discrete inverse scattering problem in one dimension
View Description Hide DescriptionA discrete version of the inverse scattering problem in one dimension is considered. While the natural formulation is somewhat different from the three‐dimensional problem with spherical symmetry, the equations of solution turn out to be almost identical. Indeed, in the continuous limit (Schrödinger equation) even the slight differences disappear. Two equivalent treatments corresponding to considering incidence from left or right are given. For actual computation a combination of the two seems most efficient.

Local observations of geodesics in the extended Kerr manifold
View Description Hide DescriptionGeodesics along the axis of symmetry in Carter's extension of the Kerr metric are divided into two types by the sign of the constant of the motion associated with the timelike Killing vector, and it is shown that this also divides them as to their place of origin on the manifold, which contains infinitely many copies of two different spaces which are flat at r = ± ∞. It is shown that geodesics cannot cross from one space to the other, but that a trajectory with properly applied acceleration can cross over.

Multiple time scales and the φ^{4} model of quantum field theory
View Description Hide DescriptionMultiple time scales perturbation theory is applied to the weakly nonlinear φ^{4}quantum field theorymodel. The multiple time scales perturbation equations are solved to lowest order, leading to the removal of secular and quasisecular terms from the standard perturbative solution. This removal occurs in a manner similar to that developed in a previous quasisecular perturbative approach which focused on small energy denominators. The multiple time scales approach provides a better rationale for the quasisecular perturbation theory, as well as providing a systematic method which can be extended to higher orders in the coupling constant. It leads to the natural introduction of a first‐order renormalized Hamiltonian, which is a well‐defined self‐adjoint operator on a certain Hilbert space of physical states. This renormalized Hamiltonian is a direct sum of Schrödinger Hamiltonians on N‐particle subspaces, which describe the interactions of pairs of particles via a nonlocal potential.

Reduction on the Lorentz subgroup of UIR's of the Poincaré group induced by a semisimple little group
View Description Hide DescriptionAll UIR's of the Poincaré group corresponding to nonzero mass are reduced on a Lorentz subgroup by means of a unique formalism. The maximal differential domain of each UIR is proved to be a nuclear space. The completeness relations (generalized matrix elements) are established between the energy‐orbital angular momentum‐total angular momentum and one of its components‐basis, and the relativistic‐orbital‐total angular momentum and one of the components of t.a.m. basis.

The general exact solution of the equation of geodesics of the general exact homogeneous plane gravitational wave
View Description Hide DescriptionThe general exact solution of the equation of geodesics for the general exact plane homogeneous gravitational wave is given by solving the vacuum field equations for the line element ds ^{2} = A (u) d x ^{2} + B (u) d y ^{2} + 2C (u) d x d y + dz ^{2} − dt ^{2}, with u = (z − t)/2^{1/2} of these waves. The subclass of the exact plane sandwich waves is given explicitly by solving exactly the flatness condition for that line element. The relative energy and momentum transfer on test particles by the sandwich wave is discussed.

Lie algebra of the hypergeometric functions
View Description Hide DescriptionIt is shown that six operators forming the Lie algebraD _{2} can be introduced in order to treat the hypergeometric functions by Lie theory techniques. The relation of these operators to the three operators of the Lie algebras l(2), which were used previously in treating the hypergeometric functions by Lie theory techniques, is discussed.

Fuzzy observables in quantum mechanics
View Description Hide DescriptionThe formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzziness inherent in actual measurement processes, relative to a multidimensional physical continuum. As an application, a position operator for the photon is defined and its intrinsic fuzziness is discussed.

Bose‐Einstein condensation in the presence of impurities. II
View Description Hide DescriptionAn alternative discussion, based on function space integration, is given of results given in our preceding paper.

A note on lattice sums in two dimensions
View Description Hide DescriptionA slight simplification of Glasser's approach for obtaining lattice sums in two dimensions is suggested. The result for the triangular lattice is given.

The evaluation of lattice sums. III. Phase modulated sums
View Description Hide DescriptionTwo‐dimensional lattice sums of the form Σ′exp(ik · s)k ^{−2n } are evaluated exactly in terms of Jacobian theta functions.

The polaron without cutoffs in two space dimensions
View Description Hide DescriptionHamiltonians for the polaron of fixed total momentum are defined using momentum cutoffs. A renormalized Hamiltonian of fixed total momentum is defined in two space dimensions by proving the strong convergence of the resolvents of the cutoff Hamiltonians. The Hamiltonian for the physical polaron is defined as the direct integral of the fixed momentum Hamiltonians.

Depolarization in nonuniform multilayered structures‐full wave solutions
View Description Hide DescriptionIn view of the recent impetus to produce rigorous solutions to more realistic models of pertinent propagation problems over a wide range of frequencies, we present in this paper full wave solutions to the problem of radio wave propagation in nonuniform multilayered structures. The electromagneticproperties of the media, the geometry of the irregular structure, and the electromagnetic source distributions are assumed to be arbitrary three‐dimensional functions of position. Generalized field transforms are employed to provide a basis for the expansion of the transverse electromagnetic fields and Maxwell's equations are reduced to a set of first‐order coupled differential equations for the forward and backward, vertically and horizontally polarized wave amplitudes. For open structures the complete wave spectrum includes the radiation term, the lateral waves, and the surface waves or trapped waveguide modes. For structures bounded by impedance walls (or perfect electric or magnetic walls μ/ε → 0 and ε/μ → 0, respectively) the fields are expressed exclusively in terms of waveguide modes. Exact boundary conditions are imposed at all the interfaces of the structure and the general solutions are not limited by the (approximate) surface impedance concept. The full wave approach employed is not restricted by frequency considerations. It is applicable to very broad classes of problems in which no single constituent of the total formal solution dominates. The full wave solutions may be applied to problems such as (i) propagation of ground waves over irregular and inhomogeneous terrain, (ii) scattering by rough surfaces and objects of finite dimensions, and (iii) propagation of guided waves in nonuniform artificial waveguides as well as in irregular ducts in the earth's crust or in the ionosphere.

On inverse scattering
View Description Hide DescriptionIn a previous paper the inverse problem associated with a hyperbolic dispersive partial differential equation with smooth coefficients was considered. The inverse problem (the determination of the coefficients) was formulated in terms of a dual set of integral equations involving measurable quantities, the kernels of the transmission, and reflection operators. These equations contained an unknown parameter which occurs in a linear manner. A better approach to determine this parameter is presented here. It involves an auxiliary equation, which is used to eliminate the unknown parameter from the integral equations. It is shown that the resulting system has a unique solution for a certain class of scattering problems. These uniqueness results are then strengthened when an additional equation is employed to reduce the dual set of integral equations to a single integral equation.

Exact formulas for 2 × n arrays of dumbbells
View Description Hide DescriptionSeveral exact results are given for the problem of enumerating arrangements of q indistinguishable dumbbells on a 2 × n array of compartments.

Physical applications of multiplicative stochastic processes. II. Derivation of the Bloch equations for magnetic relaxation
View Description Hide DescriptionThe multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.

On normalization problems of the path integral method
View Description Hide DescriptionAmbiguities of the path and of normalization in Feynman's path integral method are discussed. The investigation shows that the Feynman path integral method possesses inherent ambiguities, which can be resolved by a prescription which agrees completely with the Schrödinger equation.

On the conditions that a vector field vanish outside a given radius
View Description Hide DescriptionThe constraints that the curl and divergence of a vector field must satisfy in order that the field be identically zero outside a given radius are investigated. These result in two equations which connect the multipoles of the components of the curl and the multipoles of the divergence of the field for each given l. In an alternative proof of this constraint it is shown that the same relations still hold for the full space provided that the radial functions, which are the coefficients of the expansion of the field in vector spherical harmonics, satisfy certain conditions at infinity.

Analyticity in the coupling strength
View Description Hide DescriptionWe briefly review the methods which have been used to establish the domains of analyticity of the partial wave scattering amplitude in the presence of a Yukawa potential as a function of the coupling strength, and the methods available for proving the convergence of Padé approximants to the Neumann series of the partial wave Lippmann‐Schwinger equation. We then give a complete proof, using the Banach space technique first used by Lovelace, that the scattering amplitude is meromorphic in the coupling strength, and use Pommerenke's or Beardon's theorem to deduce the domains of convergence of the Padé approximants.

Multipole expansions and plane wave representations of the electromagnetic field
View Description Hide DescriptionA new and conceptually simple derivation is presented of the multipole expansion of an electromagnetic field that is generated by a localized, monochromatic charge‐current distribution. The derivation is obtained with the help of a generalized plane wave representation (known also as the angular spectrum representation) of the field. This representation contains both ordinary plane waves, and plane waves that decay exponentially in amplitude as the wave is propagated. The analysis reveals an intimate relationship between the generalized plane wave representation and the multipole expansion of the field and leads to a number of new results. In particular, new expressions are obtained for the electric and magnetic multipole moments in terms of certain components of the spatial Fourier transform of the transverse part of the current distribution. It is shown further that the electromagnetic field at all points outside a sphere that contains the charge‐current distribution is completely specified by the radiation pattern (i.e., by the field in the far zone). Explicit formulas are obtained for all the multipole moments in terms of the radiation pattern.

Analytic vectors for the Poincaré group in quantum field theory
View Description Hide DescriptionWe give a new proof of the theorem stating that in a quantum field theory with tempered field operators the dense domain of the polynomialalgebra of these field operators applied to the vacuum state contains a dense invariant set of analytic vectors for the representation of the Poincaré group.