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## The Boundary Value Problem

A solution of the S-L d.e. for will be an eigenfunction of the regular S-L boundary value problem if and only if the corresponding phase, obtained from the Prüfer d.e.

satisfies the corresponding end point conditions

with and .

Note that any for which these endpoint conditions hold is an eigenvalue of the regular S-L problem, and conversely, that an eigenvalue of this S-L problem will yield a phase function whenever it satisfies the required end point conditions for some .

The question now is: Does there exist a which guarantees that the two end conditions are satisfied for every ?

Ulrich Gerlach 2010-12-09