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Sturm-Liouville Problem: Prufer Equation

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Consider the Sturm-Liouville problem (pu') + Vu = 0 for the function u(x), with p(x) > 0 and V(x) = q(x) + lambda p(x).

(a) Perform the Prufer substitution u - r sin theta and u'p = r cos theta and obtain the Prufer equations for the amplitude r(x) and the phase theta (x):

r' - 1/2 ((1/p) - v) r sin 2 theta, theta' = (1/p)cos^2 theta + V sin^2 theta.

(b) Show that the phase theta is an increasing function at each zero of u(x).

(c) Show that, if V(x) > 0, u(x) has exactly one maximum/minimum between two consecutive zeros.

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Solution Summary

This shows how to perform a prufer substitution and find prufer equations, show that the phase is an increasing function, and show that there is exactly one maximum/minimum between consecutive zeroes.