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We construct the spectral expansion for the one-dimensional Schrödinger operatorwhere q x is a 1-periodic, Lebesgue integrable on [0,1], and complex-valued potential. We obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator L t , for t = 0, π, generated by this operation in L 2 0 1 and the t-periodic boundary conditions. Using it, we prove that the eigenfunctions and associated functions of L t form a Riesz basis in L 2 0 1 for t = 0, π. Then we find the spectral expansion for the operator L.  2002 Elsevier Science

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We obtain asymptotic formulas with arbitrary order of accuracy for the eigenvalues and eigenfunctions of a nonselfadjoint ordinary differential operator of order n whose coefficients are Lebesgue integrable on [0, 1] and the boundary conditions are strongly regular. The orders of asymptotic formulas are independent of smoothness of the coefficients.

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