Numerical Aspects of Weyl’s Theory

  • Michael Hehenberger


$$X\left( {\lambda ;x} \right) = \varphi \left( {\lambda ;x} \right) + m\psi \left( {\lambda ;x} \right),$$
constructed from two linearly independent initial solutions φ and ψ of the second order differential equation
$$\left\{ { - \frac{{{d^2}}}{{d{x^2}}} + q\left( x \right)} \right\}\,u\left( {\lambda ;x} \right) = \lambda u\left( {\lambda ;x} \right)$$
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.


Order Differential Equation Logarithmic Derivative Numerical Aspect Order Differential Operator Real Boundary Condition 
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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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