We obtain the classical Ambarzumyan's theorem for the Sturm-Liouville operators L t (q) with q ∈ L 1 [0, 1] and quasi-periodic boundary conditions, t ∈ [0, 2π), when there is not any additional condition on the potential q.

We obtain the asymptotic formulas of arbitrary order for the eigenvalues and the eigenfunctions of the differential operator generated by ordinary differential equation with Lebesgue integrable complex-valued coefficients and the quasiperiodic boundary conditions. Mathematics Subject Classification: 47E05

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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We consider the nonself-adjoint Sturm-Liouville operator with ∈ 1 [0, 1] and either periodic or antiperiodic boundary conditions. We obtain necessary and sufficient conditions for systems of root functions of these operators to be a Riesz basis in 2 [0, 1] in terms of the Fourier coefficients of .