Abstract: A combined-source solution is developed for electromagnetic radiation and scattering from a perfectly conducting body. In this solution a combination of electric and magnetic currents, called the combined source, is placed on the surface S of the conducting body. The combined-source operator equation is obtained from the E -field boundary-value equation. It is shown that the solution to this operator equation is unique at all frequencies. The combined-field operator equation also has a unique solution, but it is not directly applicable to the aperture radiation problem. The H -field and E -field operator equations fail to give unique solutions at frequencies corresponding to the resonant frequencies of a cavity formed by a hollow conductor of the same shape. The combined-source operator equation is solved by the method of moments. The solution, valid for a three-dimensional closed surface S , is then applied to a surface of revolution. Examples of numerical computations are given for a sphere, a cone-sphere, and a finite circular cylinder.

Abstract: We study the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular. Introduction. The problem we are considering is the parabolic evolution equation 5 u'(t)+Au(t)=O, O 3 is documented in [8] and [2]. It is shown in [8] that for p > 3, rp is in fact strongly A(0p)-stable for some 0 < Op < ir/2. For small p, Op is close to ir/2 and in the special cases p = 3, 4, rp is A-stable. Examples of rational approximations to eZ which are strongly A(O)-stable with r(oo) = 0 are provided by the family r,,(z) developed in [2]. In the second part, we investigate error estimates when the discretization in time is carried out by means of a multistep method. Zlamal gives an error bound under the assumption that the operator A is selfadjoint and the method strongly A(O)-stable. Here, error estimates are obtained if the operator A is maximal sectorial and the method strongly A(0)-stable (O < 0 < ir/2). I. Semidiscretization in Time by a Single-Step Method.

Abstract: Here A is a positive definite, linear, self-adjoint operator with domain D, in a separable Hilbert space X. The nonlinear operator M(u) has a linear domain D, satisfying 2 = D, n D, The question of existence and uniqueness for solutions of (1) goes back to Jijrgens [l], who considered a particular equation of importance in quantum field theory. Browder [2] obtained a completely operator theoretical abstract existence theorem, and his results have recently been extended by Heinz and v. Wahl [3], whose work plays an essential role in our presentation. See also the book of Reed [4]. Here we consider the approximation of solutions of (1) by the method of Faedo-Galerkin, which we rediscovered in our analysis. A detailed treatment of this method is given in the book of Lions [5], where it is used to prove the existence of weak solutions. Here, however, we consider the method from the viewpoint of the numerical and analytical approximation of (1). Our essential idea is to introduce nonlinearities M which are “reproducing” relative to a complete orthonormal sequence (C.O.S.) {ui}y, and thus obtain a finite system of explicitly known FaedoGalerkin approximating ordinary differential equations. This system can be handled by known methods of numerical integration, Lyapunov theory, etc. We show that if {t+jy are eigenfunctions of A, then the solution of the Faedo-Galerkin approximations converges to that of (1) under the assumptions of [3]. Our procedure can bc considered as a generalized separation of variables for a class of nonlinear wave equations. It extends to inhomogeneous equations U” + Au + M(u) = f; as well as parabolic equations u’ + Au + M(u) = jI The concept of a reproducing nonlinearity was first considered in [6], in the approximation of Ljusternik-Schnirelmann critical values, and was also applied to nonlinear wave equations in [7]. Solutions of the Faedo-Galerkin type were also used by Dickey [8] in the analysis of the extensible beam.

Abstract: The perturbed Hill operator\(L = - \frac{{d^2 }}{{dx^2 }} + p(x) + q(x)\), p(x+1)=p(x), q(x)∈L1(-∞,∞) is considered. An eigenfunction expansion theorem is obtained and the scattering theory is constructed. A new spectral variable, the so-called quasimomentum is introduced. The Riemann surface of the quasimomentum is constructed and investigated.

Abstract: Analytical evaluation of the double folding potential for inelastic scattering problem between nuclei is discussed. Assuming the wave functions of scattering nuclei to he de­ scribed by the SU, shell model ones, we can express the double folding potential in operator form exactly, which includes the Q·Q type term as a dominant coupling potential. Spin-spin or isospin-isospin terms coming from the folding of two-body central force is also treated by introducing a notion of effective internal wave functions of scattering nuclei. By using the operator form for the folding potential, we discuss the dynamical relation between the cluster model and the defonned potential model. § I. Introduction Recently reports are accumulating on the success£ ul applications of the folding potential in describing the collision of heavy ions.ll. 3l.Bl At the same time, in the wide region of light nuclei, cluster model using the double folding potential for inter-cluster interaction has proved to be quite successful for the comprehensive understanding of the structure of light nuclei. 2l The folding potentials are usually calculated for the diagonal parts between channels and reports on the evaluation of the coupling folding potentials are also accumulating.l).sl.Bl In view of the importance of the coupling of inelastic channels m many systems, the investigation of the coupling folding potentials is desirable to be pursued further. The purpose of this paper 1s to show that when we use the SU3 shell model wave functions for the description of scattering nuclei, the folding potential can be analytically expressed in operator form exactly. The coupling potential is found to consist mainly of the Q-Q type term. For the study of the characteristic feature of the folding potential, this analytical expression in operator form is very useful. Although our argument is valid also for the folding process of non-central two­ body force, we discuss in this paper only the folding potentials built on the central two-body forces, leaving the treatment of non-central forces in a separate paper. By using this expression of the folding potential, we can discuss the relation be­ t ween the cluster model and the deformed potential model from the dynamical

Abstract: The MINDO/3 semi-empirical SCF MO method has been extended to the generalized coupling operator (GCO) treatment of open shell systems proposed by Hirao and Nakatsuji and has been used to calculate heats of formation and equilibrium geometries of various ground state radicals and the lowest singlet and triplet excited states of a series of closed shell ground state molecules. The results are compared with those from the “half-electron”(HE) method. While the heats of formation calculated by these methods are slightly different, the predicted equilibrium geometries do not differ appreciably. The generalized coupling operator method required more computing time in nearly all cases.

Abstract: The utility of modifying the virtual orbitals of the Fock operator by introducition of an additional potential is discussed. A particularly convenient form for computational implementations is obtained, and improved methods for the practical solution of the secular problem are recommended.

Abstract: We give a formulation of the problem of propagation of scalar waves over a random surface. By a judicious choice of variables we are able to show that this situation is equivalent to propagation of these waves through a medium of random fluctuations with fluctuating source and receiver. The wave equation in the new coordinates has an additional term, the fluctuation operator, which depends on derivatives of the surface in space and time. An expansion in the fluctuation operator is given which guarantees the desired boundary conditions at every order. We treat both the cases where the surface is time dependent, such as the sea or surface, or fixed in time. Also discussed is the situation where the source and receiver lie between the random surface and another, possibly also random, surface. In detail we consider acoustic waves for which the surfaces are pressure release. The method is directly applicable to electromagnetic waves and other boundary conditions.

Abstract: Analytic expressions are derived for the hybrid-type two-centre matrix elements of the one-electron dipolar coupling operator between Slater atomic orbitals. The method used is of wide applicability. The matrix elements of interest are written in terms of a generalized form of overlap integral. The overlap integrals are evaluated using the convolution theorem/contour integration technique. Computational results are discussed and times given.

Abstract: Introducion and preliminary results In this paper we shall present some results connected with the uniform continuity of best approximations. In the last fifteen years problems dealing with local continuity of best approximation have been widely investigated. It was proved ([1], [2]) that the metric projection operator onto a finite-dimensional ~ebysev subspace of C[a, b] is pointwise Lip 1. An analogous result was obtained for the space Lp, 2 0 and Me= C[a, b] we can introduce the uniform modulus of continuity of the operator of best approximation on M as

Abstract: We present a semiclassical method for calculating the potential energy of a heavy quark-antiquark pair. Our method preserves the operator charge structure of the quark and antiquark. The operator structure of the gluon fields is approximately maintained by truncating the gluon degrees of freedom to a minimal set, a set which preserves the operator charge structure of the quark-antiquark-gluon system. The energy of this truncated system is determined using a variational principle. The potential thus determined accurately reproduces the results of renormalization-group improved perturbation theory up to and including effects of at least order ..cap alpha../sup 4/ ln..cap alpha...

Abstract: General expressions for the crystal field parameters are established by the introduction of an electronic charge density which depends on the operator O(n) = |ψ n > is an eigen...