Abstract. In this paper, we use the rotation number approach to study in detail the characteristic values of Hill's equations with two-step periodic potentials. As a result, the global structure of resonance pockets is described completely. The results in this paper show that resonance pockets behave in a sensible and fairly rich way even in this simplest case.
Based on the characterization of periodic eigenvalues using rotation numbers, we
analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger
operators with certain step potentials. This gives counter-examples to the Alikakos–Fusco
conjecture on the second periodic eigenvalues. Using this simple model, we
can also construct infinitely many resonance pockets, which are much like calabashes
emanating from a cane, of one-parameter Hill's equations.
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