scispace - formally typeset
Search or ask a question

Showing papers on "Elliptic coordinate system published in 1996"


Journal ArticleDOI
TL;DR: In this paper, the authors examined the basis functions for quantum and classical systems in two dimensions which admit separation of variables in at least two coordinate systems and showed that all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros.
Abstract: In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two‐dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n‐dimensional isotropic quantum oscillator.

121 citations


Journal ArticleDOI
TL;DR: In this paper, the Coulomb wavefunctions for the one-Coulomb-centre problem in the prolate spheroidal coordinate system were derived when the distances between the foci of the sphroidal system R are large.
Abstract: The one-Coulomb-centre problem is considered in the prolate spheroidal coordinate system. The asymptotic expansions for the separation constant and the Coulomb spheroidal quasiradial and quasiangular wavefunctions are derived when the distances between the foci of the spheroidal system R are large. The constructed wavefunctions are in excellent agreement with the exact wavefunctions in intra-atomic space, when the condition is fulfilled. The formulae obtained in the present paper can easily be generalized for the case of the two-Coulomb-centre problem.

11 citations


Journal ArticleDOI
TL;DR: In this article, integral relations between the eigenfunctions (Bessel functions and Legendre functions) of the Helmholtz equation in spherical, cylindrical and Cartesian coordinate systems are obtained.
Abstract: New integral relations between the eigenfunctions (Bessel functions and Legendre functions) of the Helmholtz equation in spherical, cylindrical and Cartesian coordinate systems are obtained. Expansion of particular solutions in one coordinate system with respect to eigenfunctions of the same equation in another coordinate system is applied. A stationary phase method is used to find the expansion coefficients.

2 citations



Journal ArticleDOI
TL;DR: In this paper, the authors consider magnetic fields lying on coordinate surfaces of an orthogonal curvilinear coordinate system and conclude that only fields on parallel planes or spherical shells can be expressed in the form provided by Low in 1980s.
Abstract: To seek nonlinear solutions of force-free magnetic fields, some symmetries or approximations are usually invoked. We consider magnetic fields lying on coordinate surfaces of an orthogonal curvilinear coordinate system. We conclude that only fields on parallel planes or spherical shells can be expressed in the form provided by Low in 1980s. These force-free fields are stable against small perturbations with rigid boundaries. Fields on cylindrical shells are also considered.

2 citations


Journal ArticleDOI
TL;DR: In this paper, the vibration-rotational kinetic energy operators of four-particle system in various coordinates were derived using a new and simple angular momentum method, respectively suitable for studying the systems described by scattering coordinate, valence coordinate, Radau coordinate, radau/Jacobi and Jacobi/valence hybrid coordinates and so on.
Abstract: The vibration-rotational kinetic energy operators of four-particle system in various coordinates are derived using a new and simple angular momentum method. The operators are respectively suitable for studying the systems described by scattering coordinate, valence coordinate, Radau coordinate, Radau/Jacobi and Jacobi/valence hybrid coordinates and so on. Certain properties of these operators and their possible applications are discussed.

2 citations



Journal ArticleDOI
TL;DR: In this paper, internal stresses of an elliptical ring sector with the cross section of a multi-connected region composed of two confocal ellipses, subjected to pure bending are analyzed.
Abstract: In this study, internal stresses of an elliptical ring sector with the cross section of a multi connected region composed of two confocal ellipses, subjected to pure bending are analyzed. Gohner's method is used for analysis and therefore, some difficulties caused by elliptical coordinates are eliminated. The analysis is limited to determining the first correction to the initial stress state for pure bending of an elliptical ring sector with the cross section of two confocal ellipses.

Book ChapterDOI
01 Jan 1996
TL;DR: In this paper, the authors developed a procedure to identify elasticity coefficients for materials that have no more symmetry planes, which is necessary to characterize a medium by ultrasonic techniques, but this procedure requires the assumption of orthorhombic symmetry and knowledge of the material symmetry axes.
Abstract: The hypothesis of an orthorhombic symmetry and the knowledge of the material symmetry axes is usually necessary to characterize a medium by ultrasonic techniques [1–5]. However a wrong setting up of the sample or the strata’s stacking defects in industrial composite material lead to the non-superposition between the material symmetry coordinate system and the observation coordinate system. The development of a procedure to identify elasticity coefficients for materials that have no more symmetry planes, is necessary.