Constructing resonance calabashes of Hill's equations using step potentials

Abstract

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrodinger 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.