A model for parallel one dimensional eigenvalues and eigenfunctions calculations

1998 ◽  
pp. 364-370
Author(s):  
Antonio Laganà ◽  
Gaia Grossi ◽  
Antonio Riganelli ◽  
Gianni Ferraro
Keyword(s):  
Eigenvalues And Eigenfunctions ◽  
One Dimensional

Vibration of Diatomic System in One-Dimensional Nanomaterials

2011 ◽  
Vol 55-57 ◽  
pp. 545-549
Author(s):  
Jun Lu
Keyword(s):  
Bound States ◽  
Hypergeometric Series ◽  
Basis Set ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Energy Eigenvalues ◽  
Diatomic System ◽  
Analytic Expressions ◽  
Hyperbolic Potential ◽  
The One

By means of the hypergeometric series method, the explicit expressions of energy eigenvalues and eigenfunctions of bound states for a diatomic system with a hyperbolic potential function are obtained in the one-dimensional nanomaterials. The eigenfunctions of a one-dimensional diatomic system, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.

On the First-Passage Distribution for the Envelope of a Nonstationary Narrow-Band Stochastic Process

1974 ◽  
Vol 41 (3) ◽  
pp. 793-797 ◽  
Author(s):  
W. C. Lennox ◽  
D. A. Fraser
Keyword(s):  
Stochastic Process ◽  
Hypergeometric Functions ◽  
Narrow Band ◽  
First Passage Time ◽  
Wide Band ◽  
Passage Time ◽  
First Passage ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Backward Equation

A narrow-band stochastic process is obtained by exciting a lightly damped linear oscillator by wide-band stationary noise. The equation describing the envelope of the process is replaced, in an asymptotic sense, by a one-dimensional Markov process and the modified Kolmogorov (backward) equation describing the first-passage distribution function is solved exactly using classical methods by reducing the problem to that of finding the related eigenvalues and eigenfunctions; in this case degenerate hypergeometric functions. If the exciting process is white noise, the analysis is exact. The method also yields reasonable approximations for the first-passage time of the actual narrow-band process for either a one-sided or a symmetric two-sided barrier.

scholarly journals ONE-DIMENSIONAL QUASI-RELATIVISTIC PARTICLE IN THE BOX

2013 ◽  
Vol 25 (08) ◽  
pp. 1350014 ◽  
Author(s):  
KAMIL KALETA ◽  
MATEUSZ KWAŚNICKI ◽  
JACEK MAŁECKI
Keyword(s):  
Relativistic Particle ◽  
Massive Particle ◽  
Relativistic Model ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Special Cases ◽  
Klein Gordon ◽  
Square Well ◽  
The One ◽  
Root Operator

The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ2c2d2/dx2 + m2c4)1/2 + V well (x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V well (x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λn are given, uniformly in n, ℏ, m, c and a, with error less than C1ℏca-1 exp (-C2ℏ-1mca)n-1. Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L2 and L∞ properties of eigenfunctions are studied.

scholarly journals Normal modes of a laterally heterogeneous body: A one-dimensional example

1978 ◽  
Vol 68 (1) ◽  
pp. 103-116
Author(s):  
Robert J. Geller ◽  
Seth Stein
Keyword(s):  
Perturbation Theory ◽  
Variational Method ◽  
Normal Modes ◽  
Second Order ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Transient Solution ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Second Order Perturbation Theory

abstract Various methods, including first- and second-order perturbation theory and variational methods have been proposed for calculating the normal modes of a laterally heterogeneous earth. In this paper, we test all three of these methods for a simple one-dimensional example for which the exact solution is available: an initially homogeneous “string” in which the density and stiffness are increased in one half and decreased in the other by equal amounts. It is found that first-order perturbation theory (as commonly applied in seismology) yields only the eigenvalues and eigenfunctions for a string with the average elastic properties; second-order perturbation theory is worse, because the eigenfunction is assumed to be the original eigenfunction plus small correction terms, but actually may be almost completely different. The variational method (Rayleigh-Ritz), using the unperturbed modes as trial functions, succeeds in giving correct eigenvalues and eigenfunctions even for modes of high-order number. For the example problem only the variational solution correctly yields the transient solution for excitation by a point force, including correct amplitudes for waves reflected by and transmitted through the discontinuity. Our result suggests but does not demonstrate, that the variational method may be the most appropriate method for finding the normal modes of a laterally heterogeneous earth model, particularly if the transient solution is desired.

scholarly journals Exact eigenvalues and eigenfunctions for a one-dimensional Gel’fand problem

2019 ◽  
Vol 60 (2) ◽  
pp. 021506
Author(s):  
Yasuhito Miyamoto ◽  
Tohru Wakasa
Keyword(s):  
Eigenvalues And Eigenfunctions ◽  
One Dimensional
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