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 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.

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

“…When p = 2, the eigenvalues λ n (q) and λ n (q) have been studied in detail so that the global structure of resonance pockets of Hill's equations can be given [9,25]. For general p, equation (2.8) reads as…”

Resonance Pockets of Hill's Equations with Two-Step Potentials

“…However, when general families of two-step potentials p ε (t) = p b1(ε),b2(ε),t1(ε) (t) are considered (which depend on ε in a nonlinear way), some resonance regions R n of (4.13) may contain infinitely many resonance pockets. One example presenting infinitely many resonance pockets inside R 2 is given in [14]. In fact, one can use (3.15), (3.16), and (3.18) to give a global description to all resonance pockets inside all resonance regions R n of (4.13).…”

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

“…Thus, we will adopt the rotation number approach to the spectrum of (1.5) [15,19]. This approach is more geometrical and is very useful in many problems [12,19,26]. It has been partially generalized to the periodic and antiperiodic eigenvalues of the one-dimensional p-Laplacian with periodic potentials [25].…”

“…Due to the asymmetry in (1.3), we will not consider in this paper the Fucˇik spectrum of (1.3)+(A) because it is more complicated than the periodic case. In Section 2, we follow the idea in [12,25,26] to introduce a rotation number function rðl þ ; l À Þ for Eq. (1.3) and the properties of rðl þ ; l À Þ are discussed.…”