Expressions For Eigenfunctions Research Articles

Comparison of the effect of suction-injection-combination on Rayleigh–Bénard convection in the case of asymmetric boundaries with those of symmetric ones

The effect of suction-injection-combination (SIC) on the linear and weakly nonlinear stability of Rayleigh–Bénard convection is considered in the paper for the cases of symmetric and asymmetric boundary conditions. Using the Maclaurin series with an appropriate number of terms, expression for eigenfunctions is obtained. The linear theory corroborates the results obtained using the chosen eigenfunctions in the limiting case of the no-SIC effect by matching accurately with the exact values concerning the critical Rayleigh number (Rac) and the wave number (αc). It is found that the effect of SIC is to stabilize the system in the case of symmetric boundaries irrespective of SIC being pro-gravity or anti-gravity. However, the effect of SIC is to stabilize/destabilize the system depending on SIC being pro-gravity or anti-gravity in the case of the asymmetric boundaries. We also noted a similar effect in the case of αc wherein a maximum error of order 10−4 was observed. The main novelty of the present work is studying the influence of SIC on the nonlinear dynamics of the considered problem. It is shown that the effect of SIC is to hasten the onset of chaos. Using various indicators (the largest Lyapunov exponent, the time series solution, the amplitude spectrum, and the phase-space plots), the dynamical behavior of the system is analyzed and the influence of SIC on the dynamics is recorded. The change due to the boundary effect and the SIC on the size of convection rolls and the trapping region where the dynamical system evolves within a bound is highlighted in the paper.

Spectral problem for the mean field Hamiltonian

We consider the mean field Hamiltonian HV = κ ΔV + ξ(·) in l2(V ), where V = {x} is a finite set. Characteristic equations for eigenvalues and expressions for eigenfunctions of HV are obtained. Using this result, the spectral representation of the solution of the corresponding ("head transition'') differential equation is derived.

Special function representations of the Poisson kernel on hyperbolic spaces

In this paper we compute explicit formulae for the Poisson kernels on the hyperbolic upper half-space $$\mathbf {H}^{n}$$ and the Poincare unit ball $$\mathbf {D}^{n}$$ . We first construct an associated Legendre function expression for eigenfunctions of the Laplacian and use superposition principle to get a solution for the Laplace equation on $$\mathbf {H}^{n}$$ . The Poisson kernel on $$\mathbf {D}^{n}$$ is obtained from that on $$\mathbf {H}^{n}$$ by letting the hyperbolic distance $$\rho =d(w,w')$$ $$(w,w'\in \mathbf {H}^{n})$$ tend to infinity. These Poisson kernels, apart from being interesting in their own right lead to various identities that seem to be novel in the context of special functions.

Calculation of eigenfunctions for imaging two-lens system with axial symmetry

Eigenfunctions of the optical operator describing limited imaging system with two lenses are considered under axial symmetry. Resulting functions are analogous to the generalized spheroidal functions which are eigenfunctions of the zero-order Hankel transformation. The expression for eigenfunctions' calculation is obtained using operator representation of the optical system. The influence of the spectrum width on the number of significant eigenvalues is investigated. Decomposition of circular, annular, and Gaussian beam by eigenfunctions is calculated and deviation from the original signals is estimated.

Branching of solutions for the problem of mean-square approximation of a real finite function of two variables by the modulus of double Fourier transformation

We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya, 51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental equations for finding the points of their branching. We also present, in the first approximation, the analytical representations of complex solutions branched-off from the real solution for the two-dimensional case of branching.

On Baxter $${\mathcal Q}$$ -Operators and their Arithmetic Implications

We consider Baxter \({\mathcal Q}\) -operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter \({\mathcal Q}\) -operators with an Archimedean version of the Polya–Hilbert operator proposed by Berry-Keating. We also elucidate the Dorey–Tateo spectral interpretation of eigenvalues of \({\mathcal Q}\) -operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman–Shalika–Shintani formula for Whittaker function in terms of characters.

Numerical simulation of spherical Couette flow

A spectral method is used to study numerically flow between two concentric rotating spheres. Analytical expressions of eigenfunctions of a Stokes operator in the spherical gap region are presented, numerical experiments are carried out by applying these expressions to the spectral approximation of spherical Couette flow, and numerical results are given.

Bound particle with magnetic moment in a space with topological defect

We consider a bounded quantum mechanical particle with spin −1/2 and a gyromagnetic ratio g, which is placed in a uniform magnetic field, in a space with a linear topological defect. We obtain the exact expressions for eigenfunctions and eigenvalues, using the approach of the continuum theory of defects, and show the dependence on the topological parameters and potential harmonic. Besides, we study the limits case and obtained the results described in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Analysis of the eigenvalue equation of the FEL amplifier with axisymmetric electron beam and diaphragm focusing line

Abstract The paper presents analysis of the eigenvalue problem of the FEL amplifier with axisymmetric electron beam and diaphragm focusing line. This novel type of an FEL amplifier has perspective to be used for applications where high average and peak radiation power is required [E.L. Saldin et al., Nucl. Instr. and meth. A 361 (1995) 317; these Proceedings, Nucl. Instr. and Meth. 375 (1996) 389; Nucl. Instr. and Meth. A 361 (1995) 101]. An FEL model is discussed wherein diffraction effects, space charge fields and energy spread of electrons in the beam are taken into account. To take into account diffraction effects at the diaphragms we apply the rigorous impedance boundary conditions proposed by Veinstein. The rigorous solutions of the eigenvalue problem have been found for the stepped electron beam profile. Analytical expressions for eigenfunctions of active open waveguide and formulae of their expansion in the eigenfunctions of passive open waveguide, are derived, too. Asymptotic behaviour of the obtained solutions is studied in details.

Dynamical stability of two-dimensional accretion flows - an analytic study

The eigenvalue problem is solved for nonaxisymmetric normal modes (the «Papaloizou-Pringle» instability) in a two-dimensional shearing flow with steady accretion. Asymptotic techniques are used to derive (analytically) explicit expressions for eigenfunctions of the unstable modes, and their growth rate. These expressions allow us to demonstrate the stabilizing effect of transonic accretion and to identify the physical cause for this stabilization