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

We consider the Hill operatorLy=−y″+v(x)y,0⩽x⩽π, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the formv(x)=ae−2irx+be2isx,a,b≠0,r,s∈N,r≠s.It is shown that the system of root functions does not contain a basis in L2([0,π],C)L2([0,π],C) if bc are periodic or if bc   are antiperiodic and r,sr,s are odd or r=1r=1 and s⩾3s⩾3.