Comment on ‘‘Exact eigenvalue equation for a finite and infinite collection of muffin-tin potentials’’

1990 ◽  
Vol 41 (14) ◽  
pp. 10224-10225 ◽  
Author(s):  
A. Gonis ◽  
R. Zeller ◽  
P. H. Dederichs ◽  
J. S. Faulkner ◽  
B. L. Györffy ◽  
...  
Keyword(s):  
Eigenvalue Equation ◽  
Infinite Collection ◽  
Exact Eigenvalue

scholarly journals Kinetic extraordinary-mode eigenvalue equation for relativistic nonneutral electron flow in planar geometry

1989 ◽  
Vol 7 (1) ◽  
pp. 55-84 ◽  
Author(s):  
Ronald C. Davidson ◽  
Han S. Uhm
Keyword(s):  
Electron Flow ◽  
Low Frequency ◽  
Companion Paper ◽  
Field Configuration ◽  
Outer Edge ◽  
Eigenvalue Equation ◽  
Planar Geometry ◽  
Canonical Momentum ◽  
Stability Properties ◽  
Long Wavelength

Use is made of the Vlasov–Maxwell equations to derive an eigenvalue equation describing the extraordinary–mode stability properties of relativistic, non-neutral electron flow in high-voltage diodes. The analysis is based on well-established theoretical techniques developed in basic studies of the kinetic equilibrium and stability properties of nonneutral plasmas characterized by intense self fields. The formal eigenvalue equation is derived for extraordinary-mode flute perturbations in a planar diode. As a specific example, perturbations are considered about the choice of self-consistent Vlasov equilibrium , where . is the electron density at the cathode (x = 0), H is the energy, and Py is the canonical momentum in the Y-direction (the direction of the equilibrium electron flow). As a limiting case, the planar eigenvalue equation is further simplified for low-frequency long-wavelength perturbations with |ω − kvd, ≪ ωυ where and and ⋯c = eB0/mc, and B0ệz is the applied magnetic field in the vacuum region xb < x ≤ d. Here, the outer edge of the electron layer is located at x = xb; ω is complex oscillation frequency; k is the wavenumber in the y-direction; ωυ is the characteristic betatron frequency for oscillations in the x′-orbit about the equilibrium value x′ = x0 = xb/2; and Vd is the average electron flow velocity in the y-direction at x = x0. In simplifying the orbit integrals, a model is adopted in which the eigenfunction approximated by , where x′(t′) is the x′-orbit in the equilibrium field configuration. A detailed analysis of the resulting eigenvalue equation for , derived for low-frequency long-wavelength perturbations, is the subject of a companion paper.

Numerical determination of the Titchmarsh-Weyl m -coefficient and its applications to HELP inequalities

1989 ◽  
Vol 426 (1870) ◽  
pp. 167-188 ◽  
Keyword(s):  
Numerical Methods ◽  
Integral Inequality ◽  
Best Constants ◽  
Eigenvalue Equation ◽  
Numerical Determination

This paper is concerned with numerical methods for finding m(λ) , the Titchmarsh-Weyl m -coefficient, for the singular eigenvalue equation - y" + qy = λy on [0, ∞) and the results are applied to the problem of finding best constants for Everitt’s extension to the Hardy-Little-wood-Pόlya (HELP) integral inequality.

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