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Dirac Systems with Discrete Spectra

Published online by Cambridge University Press:  20 November 2018

D. B. Hinton  and
J. K. Shaw
D. B. Hinton
Affiliation:
University of Tennessee, Knoxville, Tennessee
J. K. Shaw
Affiliation:
Virginia Tech, Blacksburg, Virginia
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In this paper we consider the one dimensional Dirac system

1.1

where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.

Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 1 , 01 February 1987 , pp. 100 - 122
Copyright
Copyright © Canadian Mathematical Society 1987

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