Weyl's Method Research Articles

On classification of singular matrix difference equations of mixed order

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

Near-field polarization of a high-refractive-index dielectric nanosphere on a dielectric substrate

The near-field polarization structure of a high-refractive-index dielectric nanosphere in a nonmagnetic and nonabsorbing medium is studied numerically in free space and in the vicinity of a flat dielectric substrate. Polarization patterns of the near field are obtained for linear and circular polarizations of the incident light and visualized using the polarization degree and three-dimensional Stokes parameters. Calculations are performed using the finite-element method and verified analytically using the Mie solution for scattering by a sphere with an extension of Weyl's method allowing description of reflection by a flat substrate. It is shown that the nanosphere drastically affects the polarization leading to a complex butterflylike structure. For instance, the near-field polarization is found to be qualitatively modified near the Mie electric and magnetic dipole and quadrupole resonances: linear polarization of the incident field results in circular polarization in some areas around the particle and vice versa. The areas of strong polarization change are found to be spatially localized in certain places around the particle making it possible to perform an experimental mapping. It is shown how a dielectric substrate affects the polarization distribution of the near-field of the particle. Also, several schemes of potential experiments to map the near-field intensity and polarization have been discussed.

New Exact Image Methods for Impedance Boundary Half-Space Green’s Function and Their Fast Multipole Expansion

Two foremost barriers to the development of fast and efficient simulation algorithms for electromagnetic problems in half-space and planar media are efficient Sommerfeld integral evaluation techniques and fast solvers. In this paper, we focus on challenging these two difficulties in the impedance boundary half-space, which is a proper model for common air-lossy media half-space. Based on the Laplace transformation, we first propose an alternative exact image representation for the Sommerfeld integral in half-space Green's function. In addition to its fast and absolute convergence property, this new exact image version does not contain any singularities and interprets the Sommerfeld integral as an integral over real image line. Furthermore, this representation allows a rigorous fast multipole expansion (FME), and thus it can be applied to the development of fast multipole solver with controllable precision. However, this exact image representation is not valid for the all Sommerfeld integrals in half-space Green's function. Therefore, a more generalized single mirror image representation, also exact and computationally efficient, is then presented. The strength of the mirror image is obtained through Weyl's method and steep descent path approach. Subsequently, an approximated FME scheme for this representation is proposed and studied via numerical examples. We find this approximated FME scheme has a higher accuracy than that of previous work. Numerical results show that the proposed work provides exact and efficient evaluations for Sommerfeld integrals in impedance boundary half-space Green's function and valuable insights into the development of fast multipole solver for half-space electromagnetic problems.

Near-field polarization distribution of Si nanoparticles near substrate

Structure of the near-field intensity and polarization distributions, the latter is described with the generalized 3D Stokes parameters, of a spherical Si subwavelength nanoparticle in a non-magnetic and non-absorbing media near a dielectric substrate has been studied in detail with the help of the Mie theory and an extension of the Weyl's method for the calculation of the reflection of dipole radiation by a flat surface. It is shown that for the nanoparticle near the substrate the interference effects due to the scattering by the nanoparticle and interaction with the substrate play an essential role. We also demonstrate how these effects depend on the dielectric properties of the nanoparticle, its size, distance to the substrate as well as on the polarization, wavelength and incident angle of the external light field.

Shortcuts to high symmetry solutions in gravitational theories

We apply the Weyl method, as sanctioned by Palais' symmetric criticality theorems, to obtain those—highly symmetric—geometries amenable to explicit solution, in generic gravitational models and dimension. The technique consists of judiciously violating the rules of variational principles by inserting highly symmetric, and seemingly gauge fixed, metrics into the action, then varying it directly to arrive at a small number of transparent, indexless, field equations. Illustrations include spherically and axially symmetric solutions in a wide range of models beyond D = 4 Einstein theory; already at D = 4, novel results emerge such as exclusion of Schwarzschild solutions in cubic curvature models and restrictions on ‘independent’ integration parameters in quadratic ones. Another application of Weyl's method is an easy derivation of Birkhoff's theorem in systems with only tensor modes. Other uses are also suggested.

On solutions of a boundary layer equation

A boundary layer equation is considered. A series solution in the inverse powers of Reynolds number is obtained. The existence of the solution is then proved using Weyl's method.

Spherical wave functions and dyadic Green's functions for homogeneous elastic anisotropic media.

A spherical-wave-function theory of bounded homogeneous elastic anisotropic media is developed. The anisotropic elastodynamic wave equations are solved exactly by using the method of plane-wave angular spectrum expansions. The series, integral representations, and addition theorems of the spherical wave functions of the first, second, third, and fourth kind for homogeneous elastic anisotropic media (HEAM) are presented. Weyl's method for the scalar Green's functions in isotropic media is generalized to the study of the dyadic Green's functions in HEAM. The series representations of Green's functions are of the form of separated variables. These representations are well suited to imposing a boundary condition when dealing with waves in spherically layered HEAM.

Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media.

An electromagnetic wave theory of bounded homogeneous anisotropic media is developed by using the method of angular spectrum expansion. The series and integral representations of the circular cylindrical wave functions and the spherical wave functions of the first, second, third, and fourth kind for homogeneous anisotropic media are obtained. Each coefficient of the Fourier series of a circular cylindrical wave function is a one-dimensional finite range of integration and every coefficient of the spherical-harmonic-function series is a two-dimensional finite range of integration. The addition theorem of wave functions for anisotropic media can be derived from that of wave functions for isotropic media. Weyl's method of deriving the scalar Green's function in isotropic media is generalized to the study of the dyadic Green's function in anisotropic media. The cold homogeneous magnetoplasma is considered as an illustrative example. For a cold homogeneous magnetoplasma, simplified series representations of wave functions and dyadic Green's functions are given. The distributional singular behavior of the dyadic Green's functions in the source region is investigated and taken into account by solving the static problem and the boundary integral equation is derived.

Non-linear electrostatics in inhomogeneous media

We derive and solve a functional equation, based on Weyl's method of orthogonal projections, which describes the electric field in a inhomogeneous medium with non-linear response. We construct, for the electric field and for the effective dielectric constant, an asymptotic expansion in the parameter of non-linearity. As an application, we study a system composed of small spherical elements with non-linear dielectric response embedded periodically in a host medium which is homogeneous and isotropic. In this case we obtain for the electric field an asymptotic expansion in the radius of the inclusions. The corresponding asymptotic series for the effective dielectric constant provides a generalization of the Clausius-Mossotti formula to the non-linear case.

Theory of light reflection from a substrate sparsely seeded with spheres: Comparison with an ellipsometric experiment

The light reflection from a substrate seeded with spherical particles is calculated in the approximation that the interaction between the particles can be neglected. In this approximation the problem reduces to that of the light scattering of one sphere on a substrate. The latter problem can be solved using Mie's solution for the scattering by a sphere in a homogeneous medium and an extension of Weyl's method for the calculation of the reflection of spherical waves by a flat surface. The method can be applied to particles with a radius of the order of the wavelength of the incident light. The theory explains in a satisfactory way the results of an ellipsometric experiment with growing spherical mercury particles on a carbon substrate.

Light scattering by a sphere on a substrate

The problem of light scattering by a sphere on a substrate is treated using Mie's solution for scattering by a sphere in a homogeneous medium and an extension of Weyl's method for the calculation of the reflection of dipole radiation by a flat surface. The developed theory can be applied to spheres with a radius of the order of the wavelength of the incident light. Particular solutions are given for the case of a perfectly conducting substrate and for the so-called static limit. In the latter case it is shown that the solution is equivalent to that obtained by Wind, Vlieger and Bedeaux for small spheres.

Eigen-values of Lipschitz kernels

AbstractAsymptotic estimates are obtained for the eigen-values of symmetric kernels which satisfy a Lipschitz condition of order α, where 0 < α ≤ 1, on a bounded region. For 0 < α < 1, the eigen-values are shown to be O(1/nα+½) for general kernels, and O(1/nα+1) for positive definite kernels. For α = 1, the eigen-values are shown to be O(1/n+3/2) in general, but only O(1/n2) when the kernel is positive definite. Examples are given to show these estimates are best possible for powers of n.The estimates for general kernels are known. In 1931, Hille and Tamarkin used infinite determinants to obtain weaker estimates in the case 0 < α < 1, but the same estimates as ours for α = 1. In 1937, Smithies sharpened the estimates for 0 < α < 1 to the ones we give by a variant of the method used by Weyl in 1912 for smooth kernels. Cochran obtained the same estimates in 1972 using Fredholm determinants. Our proof uses Weyl's method in its original form for 0 < α < 1, and in Smithies' variant for α = 1, so in all cases simplifies previous proofs.The estimates for positive definite kernels are, as far as we know, new. The proofs use the method of our recent paper on smooth positive definite kernels.

The Weyl Basis of the Unitary Group U(k)

Weyl's method for the construction of irreducible tensors of the unitary group is used to construct a basis for any irreducible representation of U(k) or GL(k) in terms of Bose creation operators. A simple way is indicated to select a complete but not over complete basis from the functions obtained. The basis obtained can be useful in nuclear or molecular calculations, as well as in some mathematical problems.

Representations of the Three-Dimensional Rotation Group in Terms of Direction and Angle of Rotation

We find the irreducible representations of the three-dimensional pure rotation group by Weyl's method, which makes use of the homomorphism of the special unitary group of order two onto the whole three-dimensional rotation group. The representations are realized, however, in terms of the angle of rotation in a specified direction and the spherical angles of the direction of the rotation rather than in terms of the familiar Euler angles. The results are then compared with those obtained by different methods and the advantages of the present technique are pointed out. We also derive the differential operators corresponding to infinitesimal rotations about the coordinate axis in terms of the new variables.

Diophantine approximation in fields of characteristic 𝑝

where x is an indeterminate and the coefficients ci all belong to a fixed finite field GF(pn). We shall discuss a number of problems of diophantine approximation related to the numbers of 4(. We first (?3) prove an analogue of Kronecker's theorem; the theorem has been proved previously by MIahler [8, p. 514]. We next define uniform distribution of sequences of numbers in 4( (see [9], also [7, chap. 8]) and prove a number of theorems similar to the theorems of Weyl's well known paper. The sum S= Ee(k(A)), extended over polynomials A eGF[p n x] of degree less than m, where 0(u) is a polynomial of degree k and e(a) is defined in (2.3) below, is studied by Weyl's method of approximation. It is found that if at least one coefficient of ?(u) -0(0) is irrational and 1 coc. The case k > p is left open; there are polynomials of degree p for which S=pnm (see (6.8) and (6.9)). For k = 2, p > 2, we make a more detailed study of the sum

A Practical Method for the Solution of Certain Problems in Quantum Mechanics by Successive Removal of Terms from the Hamiltonian by Contact Transformations of the Dynamical Variables Part II. Power Series in a Coordinate and Its Conjugate Momentum. The Anharmonic Oscillator by Perturbation Theory

The non-commutative algebra of polynomials in a coordinate and its conjugate momentum is reduced to common algebra by Weyl's method, and tables are given for facilitating its use. It is shown how the problem of the anharmonic oscillator can be solved by contact transformations, and tables are given for removing terms up to the third degree in its coordinate and momentum and for finding the modified Hamiltonian up to the fourth degree, which is as far as is ordinarily required in molecular theory. To illustrate the power of the method, the energy is computed up to terms of the eighth degree for energy containing terms to that degree (in the coordinate only), obtaining Dunham's result together with the constant terms of that order not given by him.

The Principle of Uncertainty in Weyl's System

Using Weyl's geometrical interpretation of quantum mechanics the fundamental relations of the transformation theory can be treated without making use of Dirac's $\ensuremath{\delta}$ function. The principle of uncertainty is particularly clear in this system. It is shown that in the discrete representation for a definite axis of the coordinate $q$ all Eigenwerte of the conjugate momentum $p$ are equally probable. It is also shown how in the limit of very fine subdivision the results of Heisenberg and Bohr are obtained. The whole treatment receives an easy interpretation by applying it to the problem of the diffraction of waves by a grating. Weyl's method of discussing quantities with continuous ranges of values is found to be equivalent to investigating diffraction phenomena in the limiting case of an infinitesimal grating space. The quantum condition $i(PQ\ensuremath{-}QP)=1$ is shown to be correct in a purely operational sense in spite of the fact that as a matrix equation it is untrue.

On the formation of usually convergent Fourier series

1. Series which converge expect at a set of content zero, or, using the expression very commonly adopted, series which converge usually, posses many of the properties which appertain to series which converge every-where. It becomes, therefore, of importance to device circumstances under which we can assert the consequence that a series converge in this manner. The subject has recently received considerable attention. so far as Fourier Series are concerned no result of even an approximately final character has been obtained. It may be supposed, indeed, that the result* of Jerosch and Weyl were at first so regarded, but, if we examine them closely in the light of the Riesz-Fischer theorems, which was known previously to the result of these authors, it becomes evident that they are merely equivalent to the statement that the Fourier Series of a function, whose square is summable, is changed into one which converges usually, if the typical coefficient a n and b n are divided by the sixth root of the integer n denoting their place in the series. Now it is difficult to believe that the question of the usual convergences of a Fourier Series can depended on the degree of the summability of the function with which it is associated and it is still more difficult to see how precisely the sixth root of n can have anything to do it. On the other hand Weyl's method, which itself marks an advance on that of Jerosch, does not obviously lend itself to any suitable modification which would secure a greater degree of generality in the result. The mistake is frequently made of confusing theoretical interest with pracitcal importance in the matter of a necessary and sufficient test. Tests which are only sufficient, but not necessary, are often much more convenient. Still more frequent it is convenient to work from first principles, and not to use any test at all. Instead of employing Weyl's necessary and sufficient condition that a series should converge usually, I have attacked the problem directly. The principles I have employed do not differ essentially from those already exposed in previous communication to this Society, but the generality and interest of the result obtained in the matter in hand seem to justify a further communication.