Numerical Aspects of Weyl’s Theory

  • Michael Hehenberger
Chapter

Abstract

Requiring
$$X\left( {\lambda ;x} \right) = \varphi \left( {\lambda ;x} \right) + m\psi \left( {\lambda ;x} \right),$$
(1)
constructed from two linearly independent initial solutions φ and ψ of the second order differential equation
$$\left\{ { - \frac{{{d^2}}}{{d{x^2}}} + q\left( x \right)} \right\}\,u\left( {\lambda ;x} \right) = \lambda u\left( {\lambda ;x} \right)$$
(2)
to satisfy a real boundary condition at the right endpoint x = b of the interval [a,b], Weyl /1/ found that m had to lie on a circle in the complex plane. For the singular case, b → ∞, Weyl’s circle shrinks either to a limit circle or to a limit point m(λ). For most physical examples the differential operator is of limit point case, implying the existence of a unique square-integrable solution for any complex λ = E + iE. As demonstrated by Titchmarsh /2/, Im m(λ) is proportional to the spectral density. In fact, as shown in detail by Chaudhuri and Everitt /3/, the spectrum of singular second order differential operators is completely characterized by the analytic properties of the m-function on the real axis.

Keywords

Order Differential Equation Logarithmic Derivative Numerical Aspect Order Differential Operator Real Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Michael Hehenberger
    • 1
  1. 1.Quantum Theory Project, Williamson HallUniversity of FloridaGainesvilleUSA

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