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

Abstract

We consider the Hill operator Ly = -y '' + v(x)y, 0 <= x <= pi, subject to periodic or antiperiodic boundary conditions (bc) with potentials of the form v(x) = ae(-2irx) + be(2isx), a, b not equal 0, r, s is an element of N, r not equal s. It is shown that the system of root functions does not contain a basis in L-2([0, pi], C) if bc are periodic or if bc are antiperiodic and r, s are odd or r = 1 and s >= 3.