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Publisher Summary This chapter focuses on the mountain cliff theorem. Many variational problems can be formulated in such a way that one is seeking solutions for a C' functional G defined on a real Hilbert space H. One successful method is the mountain pass lemma of Ambrosetti–Rabinowitz. The basic idea is to consider the set S of all continuous paths P connecting 0 and e hoping to find one on which the maximum of G is no greater than its maximum on any other path in S. If such a path exists, it is a true mountain pass over which a traveler can go without climbing higher than necessary. The basic idea is to consider the set S of all continuous paths P connecting 0 and e hoping to find one on which the maximum of G is no greater than its maximum on any other path in S.

We discuss a system of equations that describes many (if not most) wave propagation phenomena of classical physics. We consider spectral and scattering theory under minimal assumptions on the coefficients.