Divergence of spectral decompositions of Hill operators with two exponential term potentials

Abstract

We consider the Hill operator $$Ly=-y^{\prime \prime }+v(x)y,0\le x\le \pi ,$$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$v(x)=a{e}^{-2irx}+b{e}^{2isx},a,b\ne 0,r,s\in \mathbb{N},r\ne s.$$ It is shown that the system of root functions does not contain a basis in $L^2([0,\pi ],\mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r,s$ are odd or $r=1$ and $s\ge 3.$