We consider Schrödinger operators of the form H ϭϪ⌬ϩVϩW on L 2 (R ) ͑ϭ1, 2, or 3͒ with V periodic, W short range, and a real non-negative parameter. Then the continuous spectrum of H has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of H that are functions of the parameter . Let (a,b) be a gap and E()(a,b) an eigenvalue of H . We study the asymptotic behavior of E() as approaches a critical value 0 , called a coupling

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... t threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E()↓a as ↓ 0 , is E()Ϫaϳc(Ϫ 0 ) ␣ for some ␣Ͼ0 and c 0, or is there an expansion in some other quantity? As one expects from previous work in the case Vϭ0, the answer strongly depends on .