Mathematical Supplement

Abstract

We begin our discussion with the eigenvalue equation

$$ \hat L\psi \left( {x,L} \right) = L\psi \left( {x,L} \right), $$

((5.1))

which is supposed to have a continuous spectrum with the eigenvalues L and the eigenfunctions ψ(x,L). Now we integrate (5.1) with respect to L over the small interval ΔL, and obtain

$$ \hat L\Delta \psi (x,L) = \int\limits_L^{L + \Delta L} {L\psi } (x,L)dL, $$

((5.2))

where

$$ \Delta \psi \left( {x,L} \right) = \int\limits_L^{L + \Delta L} {\psi \left( {x,L} \right)} dL $$

((5.3))

is called the eigendifferential of the operator L̂, introduced (as mentioned earlier) by the great mathematician, H. Weyl. The eigendifferential is a special wave group which has only a finite extension in space (in the x domain), similar to the previously studied wave groups; hence, it vanishes at infinity and therefore can be seen in analogy to bound states. Now we show that indeed the functions ψ(x, L) are not orthogonal, but the eigendifferentials Δψ(x, L) are. Furthermore, because the Δψ(x, L) have finite spatial extension, they can be normalized. Then in the limit ΔL→0, a meaningful normalization of the functions ψ(x, L) themselves follows: the normalization on δ-functions.