##### Abstract

We consider Schrodinger operators of the form H-lambda= -Delta + V + lambda W on L-2(R-v) (v=1, 2, or 3) with V periodic, W short range, and lambda a real non-negative parameter. Then the continuous spectrum of H-lambda has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of H-lambda that are functions of the parameter lambda. Let (a,b) be a gap and E(lambda)E(a,b) an eigenvalue of H-lambda. We study the asymptotic behavior of E(lambda) as lambda approaches a critical value lambda(0), called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E(lambda)down arrow a as lambda down arrow lambda(0), is E(lambda)-a similar to c(lambda - lambda(0))(alpha) for some alpha>0 and c not equal 0, or is there an expansion in some other quantity? As one expects from previous work in the case V=0, the answer strongly depends on v. (C) 1998 American Institute of Physics.