  In this paper we consider a Schrodinger eigenvalue problem with a potential
consisting of a periodic part together with a compactly supported defect
potential. Such problems arise as models in condensed matter to describe color
in crystals as well as in engineering to describe optical photonic structures.
We are interested in studying the existence of point eigenvalues in gaps in the
essential spectrum, and in particular in counting the number of such
eigenvalues. We use a homotopy argument in the width of the potential to count
the eigenvalues as they are created. As a consequence of this we prove the
following significant generalization of Zheludev's theorem: the number of point
eigenvalues in a gap in the essential spectrum is exactly one for sufficiently
large gap number unless a certain Diophantine approximation problem has
solutions, in which case there exists a subsequence of gaps containing 0,1 or 2
eigenvalues. We state some conditions under which the solvability of the
Diophantine approximation problem can be established.
