Estimates for periodic and Dirichlet eigenvalues of the Schrodinger operator with singular potentials

In this paper, the periodic and the Dirichlet problems for the Schrodinger operator -(d(2)/dx(2))+V are studied for singular, complex-valued potential s V in the Sobolev space H-per(-alpha)[0, 1] (0 less than or equal to alpha < 1). The following results are shown: (1) The periodic spectrum consists of a sequence (lambda (k))(k greater tha n or equal to0) of complex eigenvalues, satisfying the asymptotics (for any epsilon > 0) l(2n-1), lambda (2) = n(2)pi (2) + (V) over cap (0) +/- root(V) over cap(-2 n) (V) over cap (2n) + O(n(3 alpha /2-1/2+epsilon)), where (V) over cap (k) denote the Fourier coefficients of V. (2) The Dirichlet spectrum consists of a sequence (mu (n))(n greater than o r equal to1) of complex eigenvalues satisfying the asymptotics (for any eps ilon > 0) mu (n) = n(2)pi (2) + <(Vover cap>(0) - (V) over cap(-2n) + (V) over cap (2 n)/2 + O(n(2 alpha -1 + epsilon)). (C) 2001 Academic Press.