Harry E. Moses

EIGENFUNCTIONS OF THE CURL OPERATOR, ROTATIONALLY INVARIANT HELMHOLTZ THEOREM, AND APPLICATIONS TO ELECTROMAGNETIC THEORY AND FLUID MECHANICS*

In the present paper we introduce eigenfunctions of the curl operator. The expansion of vector fields in terms of these eigenfunctions leads to a decomposition of such fields into three modes, one of which corresponds to an irrotational vector field and two of which correspond to rotational circularly polarized vector fields of opposite signs of polarization. Under a rotation of coordinates, the three modes which are introduced in this fashion remain invariant. Hence we have introduced the Helmholtz decomposition of vector fields in an irreducible, rotationally invariant form.These expansions enable one to handle the curl and divergence operators simply. As illustrations of the use of the curl eigenfunctions, we solve four problems. The first problem that is solved is the initial value problem of electromagnetic theory with given time- and space-dependent sources and currents and we show that the radiation and longitudinal modes uncouple in a very simple way. In the second problem we show how fluid motion...

A kernel of Gel’fand–Levitan type for the three‐dimensional Schrödinger equation

In a previous paper we introduced a Green’s function for the three‐dimensional Schrodinger equation analogous to the Green’s function used to obtain the integral equation for the Jost wave functions in one dimension. The three‐dimensional Green’s function was used to define Jost wave functions for the three‐dimensional problem and the completeness relations for these wave functions were obtained. In the present paper we use the three‐dimensional Green’s function to construct influence functions for the 3+3 ultrahyperbolic partial differential equation which have analogs to the causal properties of the corresponding influence functions for the 1 + 1 hyperbolic partial differential equation. Just as the 1 + 1 influence function can be used to obtain an integral equation for the one‐dimensional Gel’fand–Levitan kernel in terms of the scattering potential, we use the 3 + 3 influence function to obtain an analogous integral equation for our proposed Gel’fand–Levitan kernel for the three‐dimensional problem. Though much of the formalism for finding the properties of the kernel for the three‐dimensional problem can be carried out in a straightforward manner, the interpretation of the triangularity properties is more difficult than in the one‐dimensional case because of the complicated geometrical picture associated with the notion of causality. In addition to its use in obtaining a Gel’fand–Livitan kernel, the 3+3 influence function can be used to simplify the second term in an expansion of the potential in terms of the minimal scattering data. This simplification is also given. In the Appendix the asymptotic form of the three‐dimensional Jost wave function is given in a form which is analogous to the asymptotic form for the one‐dimensional Jost wave function and which is compatible with our notion of triangularity for the Gel’fand–Levitan kernel.

The time‐dependent inverse source problem for the acoustic and electromagnetic equations in the one‐ and three‐dimensional cases

The object of the time‐dependent inverse source problem of electromagnetic theory and acoustics is to find time‐dependent sources and currents, which are turned on at a given time and then off to give rise to prescribed radiation fields. In an early paper for the three‐dimensional electromagnetic case, the present writer showed that the sources and currents are not unique and gave conditions which make them so. The ideas of that paper are reformulated for the three‐dimensional electromagnetic case and extended to the acoustical three‐dimensional case and the one‐dimensional electromagnetic and acoustic cases. The one‐dimensional cases show very explicitly the nature of the ambiguity of the choice of sources and currents. This ambiguity is closely related to one which occurs in inverse scattering theory. The ambiguity in inverse scattering theory arises when one wishes to obtain the off‐shell elements of the T matrix from some of the on‐shell elements (i.e., from the corresponding elements of the scatterin...

Acoustic and electromagnetic bullets: derivation of new exact solution of the acoustic and Maxwell's equations

Previously it has been shown that all finite energy, causal solutions of the time-dependent three-dimensional acoustic equation and Maxwell’s equations have forms for large radius that, except for a factor

Gel’fand–Levitan equations with comparison measures and comparison potentials

1/r

A generalization of the Gel’fand–Levitan equation for the one‐dimensional Schrödinger equation

, represent one-dimensional wave motions along straight lines through the origin. The asymptotic region is called the wave zone. Conversely, it also has been shown how the exact finite energy causal solutions can be obtained from the asymptotic solutions in the wave zone through the use of a refined Radon transform. Thus a way of finding exact, causal three-dimensional solutions from the essentially one-dimensional solutions in the wave zone has been obtained.When the asymptotic solutions are confined to a finite radial interval within a cone, the exact solutions are termed “bullets” and asymptotically represent packets of acoustic and electromagnetic energy “shot” through the cone, which is a kind of “rifle.” No reflectors are needed. In the present paper explicit examples of such exact solutions ...

The general solution of the three-dimensional acoustic equation and of Maxwell's equations in the infinite domain in terms of the asymptotic solution in the wave zone

Using an abstract form of the Gel’fand–Levitan equation, it is shown how a solution of the equation corresponding to a given weight operator can be found in terms of a solution for the equation with a different weight operator. The resulting Gel’fand–Levitan equation is a generalization of the original one. To achieve our result, an analog of a canonical transformation for direct scattering is used. The effect of the use of the transformation is to include part of the scattering potential (the comparison potential) in the unperturbed Hamiltonian. The generalized Gel’fand–Levitan equation has the advantage that if the weight operator for a given Gel’fand–Levitan equation is close to that for an already solved Gel’fand–Levitan equation, the solution of the first can be obtained from the second by using the solution of the second as a first approximation in an iteration procedure or as a trial function in a variational procedure. The method is illustrated by considering the inverse problem for the one‐dimens...

Eigenvalues and eigenfunctions associated with the Gel’fand–Levitan equation

The Gel’fand–Levitan equation for the one‐dimensional Schrodinger equation is generalized to the case that the unperturbed Hamiltonian contains part of the scattering potential, this part being denoted by V0(x), and that the direct scattering problem has been solved for this Hamiltonian. Hence one knows the reflection coefficient b0(k), the point eigenvalues E0i, and the normalizations of the corresponding eigenfunctions C0i. We are given b1(k), E1i, C1i, which are the corresponding quantities for full potential V1(x) =V0(x)+ΔV (x). A Gel’fand–Levitan equation is set up in terms of b1(k)−b0(k) and the difference in measures for the discrete spectra for V0 and V1, respectively, from which ΔV can be found. One may regard the new algorithm as providing a means to modify a known potential to accommodate prescribed changes in the reflection coefficient and changes in the nature of the discrete spectrum. The generalization has applications to the Korteweg–de Vries equation. It is shown that a kind of ’’superpos...

Propagation of an electromagnetic field through a planar slab

The general solution of the three‐dimensional scalar wave equation (or acoustic equation) and of Maxwell’s equations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.

A Refinement of the Radon Transform and Its Inverse

It is shown here that the solutions of the Gel’fand–Levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class. The details are sketched for the case of rational reflection coefficients, and carried out for the simplest class of examples.