How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates

2005 ◽  
Vol 122 (13) ◽  
pp. 134101 ◽  
Author(s):  
Richard Dawes ◽  
Tucker Carrington
Keyword(s):  
Direct Product ◽  
Energy Levels ◽  
Basis Functions ◽  
Coupled Systems ◽  
One Dimensional ◽  
The One

Wavelets in the Solution of Nongray Radiative Heat Transfer Equation

1998 ◽  
Vol 120 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Y. Bayazitoglu ◽  
B. Y. Wang
Keyword(s):  
Transfer Equation ◽  
Solution Method ◽  
Radiative Heat ◽  
Radiative Transfer Equation ◽  
Basis Functions ◽  
Wavelet Basis ◽  
System Of Equations ◽  
Heat Transfer Equation ◽  
One Dimensional ◽  
The One

The wavelet basis functions are introduced into the radiative transfer equation in the frequency domain. The intensity of radiation is expanded in terms of Daubechies’ wrapped-around wavelet functions. It is shown that the wavelet basis approach to modeling nongrayness can be incorporated into any solution method for the equation of transfer. In this paper the resulting system of equations is solved for the one-dimensional radiative equilibrium problem using the P-N approximation.

scholarly journals A Numerical Variational Method for Calculating Accurate Vibrational Energy Separations of Small Molecules and Their Ions

1986 ◽  
Vol 39 (5) ◽  
pp. 749 ◽  
Author(s):  
G Doherty ◽  
MJ Hamilton ◽  
PG Burton ◽  
EI von Nagy-Felsobuki
Keyword(s):  
Finite Element ◽  
Variational Method ◽  
Small Molecules ◽  
Vibrational Energy ◽  
Basis Functions ◽  
Variational Procedure ◽  
One Dimensional ◽  
Trial Wavefunction ◽  
Vibrational Energies ◽  
The One

A combination of known methods have been spliced together in order to calculate accurate vibrational energies and wavefunctions. The algorithm is based on the Rayleigh-Ritz variational procedure in which the trial wavefunction is a linear combination of configuration products of one-dimensional basis functions. The Hamiltonian is that due to Carney and Porter (1976). The kernel of the algorithm consists o( the one-dimensional basis functions, which are the finite element solutions of the associated one-dimensional problems.

scholarly journals Spectral Statistics of the Triaxial Rigid Rotator: Semiclassical Origin of Their Pathological Behavior

2003 ◽  
Vol 17 (15) ◽  
pp. 803-812
Author(s):  
V. R. Manfredi ◽  
V. Penna ◽  
L. Salasnich
Keyword(s):  
Angular Momentum ◽  
Spectral Properties ◽  
Total Angular Momentum ◽  
Energy Levels ◽  
Rigid Rotator ◽  
One Dimensional ◽  
Spectral Statistics ◽  
Rotational Motions ◽  
Pathological Behavior ◽  
The One

In this paper we investigate the local and global spectral properties of the triaxial rigid rotator. We demonstrate that, for a fixed value of the total angular momentum, the energy spectrum can be divided into two sets of energy levels, whose classical analogs are librational and rotational motions. By using diagonalization, semiclassical and algebric methods, we show that the energy levels follow the anomalous spectral statistics of the one-dimensional harmonic oscillator.

scholarly journals ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL

2017 ◽  
Vol 57 (6) ◽  
pp. 385 ◽  
Author(s):  
Silvestro Fassari ◽  
Manuel Gadella ◽  
Luis Miguel Nieto ◽  
Fabio Rinaldi
Keyword(s):  
Discrete Spectrum ◽  
Energy Levels ◽  
Trace Class ◽  
Bound State ◽  
Great Accuracy ◽  
Coupling Constant ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Gaussian Potential ◽  
The One

<p>We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space, taking advantage of the unique feature of this potential, that is to say its invariance under Fourier transform. <br />Given that such integral operators are trace class, it is possible to determine the energy levels in the discrete spectrum of the Hamiltonian as functions of <span>the coupling constant with great accuracy by solving a finite number of transcendental equations. We also address the important issue of the coupling constant thresholds of the Hamiltonian, that is to say the critical values of λ for which we have the emergence of an additional bound state out of the absolutely continuous spectrum. </span></p>

scholarly journals Discrete-Time Orthogonal Spline Collocation Method for One-Dimensional Sine-Gordon Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaoquan Ding ◽  
Qing-Jiang Meng ◽  
Li-Ping Yin
Keyword(s):  
Discrete Time ◽  
Gordon Equation ◽  
Basis Functions ◽  
Spline Collocation ◽  
Time Level ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Orthogonal Spline Collocation ◽  
Collocation Scheme ◽  
The One

We present a discrete-time orthogonal spline collocation scheme for the one-dimensional sine-Gordon equation. This scheme uses Hermite basis functions to approximate the solution throughout the spatial domain on each time level. The convergence rate with orderO(h4+τ2)inL2norm and stability of the scheme are proved. Numerical results are presented and compared with analytical solutions to confirm the accuracy of the presented scheme.

The United Adaptive Learning Algorithm for the Link Weights and Shape Parameter in RBFN for Pattern Recognition

1997 ◽  
Vol 11 (06) ◽  
pp. 873-888 ◽  
Author(s):  
De-Shuang Huang
Keyword(s):  
Adaptive Learning ◽  
Shape Parameter ◽  
Learning Algorithm ◽  
Basis Functions ◽  
Recursive Least Squares ◽  
Training Method ◽  
One Dimensional ◽  
Radial Basis ◽  
Radar Targets ◽  
The One

This paper proposes a united training method of the link weights of the Gaussian radial basis function networks (GRBFN) and the shape parameter α of the RBF. The training method corresponding to the former is a kind of recursive least squares backpropagation (RLS-BP) learning algorithm which is an accurately recursive method, the training method corresponding to the latter is an adaptive gradient descending (AGD) searching algorithm which is an approximately approaching method. We use the one-dimensional images of radar targets to study the effect of the shape parameter α on the rate of recognition, and survey the changes of the shape parameter αs of radial basis functions corresponding to different hidden nodes, and present the judgement confidence curves of different radar targets. In addition, the forgotten factor λ which makes the effects on the speed of convergence is also discussed. The experimental results are presented.

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