Eigenfunctions Of Eigenvalue Problem Research Articles

Understanding the physics of eigenvalue-eigenfunction problems: Rotating beam problem

Eigenvalue-eigenfunction problems frequently appear in many physical areas. Some mathematical experience is needed to identify whether the differential system is an eigenvalue-eigenfunction problem or not. Apart from the mathematical nature of the problem, the eigenvalue-eigenfunction solutions have physical interpretations which have to be addressed properly for real problems. The rotating beam problem is treated to exploit the mathematical and physical nature of such problems and the conditions to divert from the eigenvalue-eigenfunction problem. The rotation of a beam about its symmetry axis along its length and about another axis parallel to its symmetry axis changes the nature of the problem. While the former is an eigenvalue-eigenfunction problem, the latter is not. The interpretations of the physical consequences of the solutions are discussed in detail. The problem can be used as supplementary material in undergraduate courses such as differential equations, mechanics and dynamics.

Bound States of a System of Two Fermions on Invariant Subspace

We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice Z3 with special potential . The corresponding Shrödinger operator H(k) of the system has an invariant subspac L-123(T3) , where we study the eigenvalues and eigenfunctions of its restriction H-123(k). Moreover, there are shown that H-123(k1, k2, π) has also infinitely many invariant subspaces , where the eigenvalues and eigenfunctions of eigenvalue problem are explicitly found.

Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

It is shown that any \(\mu \in \mathbb{C}\) is an infinite multiplicity eigenvalue of the Steklov smoothing operator \(S_h\) acting on the space \(L^1_{loc}(\mathbb{R})\). For \(\mu \neq 0\) the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.

The Liouville–Neumann expansion in one-dimensional boundary value problems

We consider boundary value problems of the form with f′, g and h continuous in [0, 1]. We use the Liouville–Neumann expansion to design a succession of functions y n (x) that converge uniformly on [0, 1] to the solution y(x) of that problem. In particular, when f(x), g(x) and h(x) are polynomials, y n (x) are also polynomials. We show that the Liouville–Neumann algorithm may be used to approximate eigenvalues and eigenfunctions of eigenvalue problems of the form with f′, g and σ continuous in [0, 1]. It may be also used to approximate zeros of solutions of regular second-order linear differential equations and, in particular, of some special functions.

The Completeness of Eigenfunctions of the Tidal Equation on an Equatorial Beta Plane*

Abstract The approximate tidal theory on an equatorial beta plane has been widely applied to tropical atmospheric dynamics. There are many successful examples of such applications. However, the mathematical and physical origin of the recently discovered continuous spectrum associated with meridional eigenfunctions of negative equivalent depth is yet to be given, and the completeness of the meridional eigenfunctions in the approximate tidal theory remains to be proved. In this note, a proof of the completeness of the meridional eigenfunction is presented. The differential equation is first transformed into an equivalent integral equation that relates the solution of the differential equation to the corresponding Green's function. It is then shown that the Green's function corresponding to the meridional eigenvalue–eigenfunction problem is linear, self-adjoint, completely continuous, and square integrable over the meridional infinite domain under the principle of analytic continuation. Therefore, the eigenf...

Nonhydrostatic Atmospheric Normal Modes on Beta-Planes

—To facilitate the understanding of nonhydrostatic effect in global and regional nonhydrostatic models, the normal modes of a nonhydrostatic, stratified, and compressible atmosphere are studied using Cartesian coordinates on midlatitude and equatorial β-planes. The dynamical equations without forcing and dissipation are linearized around the basic state at rest, and solved by using the method of separation of variables. An eigenvalue-eigenfunction problem is formulated, consisting of the horizontal and vertical structure equations with suitable boundary conditions. The wave frequency and the separation parameter, referred to as “equivalent height,” appear in both the horizontal and vertical characteristic equations as a coupled problem, unlike the hydrostatic case. Therefore, the nonhydrostatic equivalent height depends not only on the vertical modal scale, as in the hydrostatic case, but also on the zonal and meridional modal scales. Numerical resu lts on the dispersion relations are presented for an isothermal atmosphere. Three kinds of normal modes, namely acoustic, gravity, and Rossby modes, are solved and compared with the corresponding global solutions. Nonhydrostatic effects are studied in terms of normal modes in a wide range of wavelengths from small to planetary scales. It is demonstrated that Rossby modes are hardly affected by nonhydrostatic effects regardless of wavelengths. However, nonhydrostatic effects on gravity modes become significant for smaller horizontal and deeper vertical scales of motion. The equivalent height plays a particularly important role in evaluating nonhydrostatic effects of normal modes on the equatorial β-plane, because the equivalent height appears in the scaling of meridional distance variable of the eigenfunctions. The implementation of nonhydrostatic normal mode analysis on high-resolution numerical modeling is also discussed.

Normal Modes of a Global Nonhydrostatic Atmospheric Model

Abstract Anticipating use of a very high resolution global atmospheric model for numerical weather prediction in the future without a traditional hydrostatic assumption, this article describes a unified method to obtain the normal modes of a nonhydrostatic, compressible, and baroclinic global atmospheric model. A system of linearized equations is set up with respect to an atmosphere at rest. An eigenvalue–eigenfunction problem is formulated, consisting of horizontal and vertical structure equations with suitable boundary conditions. The wave frequency and the separation parameter, referred to as “equivalent height,” appear in both the horizontal and vertical equations. Hence, these two equations must be solved as a coupled problem. Numerical results are presented for an isothermal atmosphere. Since the solutions of the horizontal structure equation can only be obtained numerically, the coupled problem is solved by an iteration method. In the primitive-equation (hydrostatic) models, there are two kinds of ...

A C. NEUMANN SYSTEM AND THE INVOLUTIVE REPRESENTATION OF THE SOLUTION OF HIERARCHY FOR THE COUPLED NONLINEAR WAVE EQUATION

In this paper, the Lax representation for the coupled nonlinear wave equation hierarchy is obtained, and the nonlinear constraint between the potential and eigenfunctions of eigenvalue problem is introduced, a complete integrable C. Neumann system is led. Finally, as a result, the solutions of the coupled nonlinear wave equation hierarchy are obtained by using the involutive representation for the commutable flows.

On the computation of induced norms for non-compact Hankel operators arising from distributed control problems

This paper considers the problem of computing the induced L 2 norm for Hankel operators which are not necessarily compact. In the literature such a norm has traditionally been computed by solving an infinite-dimensional eigenvalue-eigenfunction problem. In this paper, it is shown that a sequence of finite-dimensional eigenvalue-eigenvector problems can be solved to estimate the desired induced norm.

Regularity and Numerical Solution of Eigenvalue Problems with Piecewise Analytic Data

This paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems. A detailed and systematic numerical study of these approximations is presented, together with an analysis of the numerical results in light of the theoretical results. The specific aim is to assess the reliability of the theoretical results, which are of an asymptotic nature, as a guide to practical computations, which may take place in the pre-asymptotic phase, and to look for characteristic features of the numerical results that are not completely explained by known theoretical results.

Application of Vertical Normal Mode Expansion to Problems of Baroclinic Instability

As an alternative to the finite difference method, we explore the use of the spectral method with normalmodes as the basis functions for discretizing dependent variables in the vertical direction in order to obtainnumerical solutions to time dependent atmospheric equations. The normal modes are free solutions to the timedependent perturbation equations linearized around the atmosphere at rest. To demonstrate the feasibility ofnormal mode representation in the spectral vertical discretization, the vertical normal mode expansion is appliedto the quasi-geostrophic potential vorticity equation to investigate the traditional baroclinic instability of Charneyand Green types on a zonal flow with a constant vertical shear. The convergence of the numerical solutions isexamined in detail in relation to the spectral resolution of expansion functions.We then extend the method of vertical normal mode expansion to solve the problem of baroclinic instabilityon the sphere. Two aspects are different from the earlier example. One is use of the primitive equations insteadof the quasi-geostrophic system and the other is application of normal mode expansions in the horizontal, aswell as vertical direction. First, we derive the evolution equations for the spectral coefficients of truncated seriesin three-dimensional normal mode functions by application of the Galerkin procedure to the global primitiveequations linearized around a basic zonal flow with vertical and meridional shear. Then, an eigenvalue-eigenfunction problem is solved to investigate the stability of perturbation motions superimposed on the 30' jetexamined earlier by Simmons, Hoskins and Frederiksen. From these two examples, it is concluded that thenormal mode spectral method is a viable numerical technique for discretizing model variables in the vertical.

Instability of Non-Zonal Baroclinic Flows: Multiple-Scale Analysis

The linear instability of a non-zonal flow can be reduced to an eigenvalue-eigenfunction problem, governed by a nonseparable partial differential equation (Niehaus, 1980). Approximate solutions, found by the method of multiple scales, are derived here and compared with earlier results found using a spectral method. The amplitude maxima are correctly located. The zonal variations of local wavenumber and of amplitude are qualitatively correct, but not sufficiently extreme. Because the method is oversensitive to local conditions, and less sensitive to global constraints, this comparison provides theoretical limits to the possibility of parameterizing transient eddies in terms of the local time mean state of the atmosphere. The method can be extended easily to flows with more realistic vertical structure.

Influence of impurities, edge effects and refuelling on the thermal stability of a burning D—T plasma

The thermal stability of a D—T plasma with a core temperature in the range from 5 keV to 20 keV is discussed. Impurities, edge effects and a neutral particle background which substitutes the plasma lost by diffusion are taken into account. Based on a linear instability analysis for the space-time-dependent energy balance in cylindrical geometry, regimes of stable and quasi-stable burning, the latter characterized by a limitation of the product of axial ion density and burning time, are determined by solving the coupled eigenvalue-eigenfunction problem for the equilibrium temperature and for the perturbation profiles.

Limit theorems for solutions of stochastic differential equation problems

In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.Limit theorems are proved for the eigenvalues and the eigenfunctions of eigenvalue problems and for the solutions of boundary value problems and initial value problems.

A direct numerical approach to three nucleon bound state problems

This paper describes a direct numerical method for solving three nucleon bound state problems. The method has been tried out and developed on a sequence of two and three nucleon problems of increasing difficulty. This work forms part of an attack on the problem of calculating the triton binding energy with the nucleon-nucleon interaction described by the Hamada-Johnston potential. The method uses the analysis of Derrick and Blatt which reduces Schrödinger's equation for the ground state of the triton to an eigenvalue-eigenfunction problem consisting of a set of linked partial differential equations in three variables. In the main calculation described the full three nucleon ground state problem is simplified in two respects. Firstly, the spin orbit forces are omitted from the potential leaving only central and tensor forces. Secondly, in the expansion of the ground state wavefunction in terms of states of definite total angular momentum, iso-spin, and parity, only nine of the 16 components are included resulting in a problem involving nine partial-differential equations. A direct numerical approach is employed in which functions of three variables are represented by three-dimensional tables of numbers and derivatives by finite-difference operators. The numerical problem is thus obtained in the form of an eigenvalue-eigenvector problem. It is solved on a sequence of mesh sizes and an approximation to the analytic binding energy obtained by extrapolating to zero mesh size.

The Semidiurnal Oscillation in a Thermalgeostrophic Atmosphere

Abstract The problem of atmospheric response to tidal oscillations is re-examined for consistent and fairly general atmospheric models. An approximate extension of the separation method is developed, with an adjusting mixed factor, inherently defined through and defining a residual eigenvalue-eigenfunction problem consisting of one formally unchanged radial equation and a set of modified angular equations. The possibility of a considerably amplified S22 reappears, and the appropriate “tropospheric” first mode fits observations surprisingly well. A seasonal dependent eigensolution, equatorially asymmetric at the solstices, approaching cos3ϕ at the surface, but decreasing much faster at the tropopause, and an almost constant significantly increased equivalent depth h2,22 implying increased response and radial exponentiality, are essential features of this new S2,22. The set of angular equations incorporates a kind of nonlinearity. Although a tropospherically excited semidiurnal mode may be strongly amplifie...