On algebraic multiplicity of (anti)periodic eigenvalues of Hill’s equations

Abstract

We construct two explicit examples of Hill’s equations with complex-valued potentials such that the algebraic multiplicity of some (anti)periodic eigenvalue E E equals 1 + 2 p i 1+2p_{i} with p i ≥ 1 p_{i}\geq 1 , where p i p_{i} denotes the immovable part of E E as a Dirichlet eigenvalue. These examples confirm a phenomena about Hill’s equations in (Gesztesy and Weikard, Acta Math. 176 (1996), 73–107).