Phase Analysis of a Sturm-Liouville System

Every Sturm-Liouville system has a personality, which is encoded in its phase. In other words, the phase is the brains of the regular Sturm-Liouville system.

It is the phase which determines where a given solution has a maximum.

It is the phase which determines where a given solution is zero.

It is the phase which determines where a given solution oscillates.

It is the phase which determines how many zeroes a given solution has in its domain of definition.

Thus, when one thinks of the questions, ``How do the boundary and the associated eigenvalue parameter $\lambda$ control the nature of the solution to the regular Sturm-Liouville problem?'' one should actually ask a more penetrating question:

``How do the boundary conditions and the associated eigenvalue parameter $\lambda$ control the phase of the solution to the S-L problem?''

The existence of allowed (eigen)values of $\lambda$ and the concomitant eigenfunction is determined entirely by the phase. Let us, therefore, recast the mixed D-N boundary conditions in terms of this phase.