Counting defect modes in periodic eigenvalue problems

In this paper we consider a one-dimensional Schrödinger 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 arecreated, essentially by computing a Maslov index. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in theessential spectrum is exactly 1 for a sufficiently large gap number unless a certain Diophantineapproximation problem has solutions, in which case there exists a subsequence of gaps containing0, 1, or 2 eigenvalues. We state some conditions under which the solvability of the Diophantineapproximation problem can be established.