Abstract
We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and the quasiperiodic boundary conditions. Using these asymptotic formulas, we find conditions on the coefficients for which the root functions of this operator form a Riesz basis. Then, we obtain the uniformly convergent spectral expansion of the differential operators with the periodic matrix coefficients.
Highlights
Let L P2, P3, . . . , Pn ≡ L be the differential operator generated in the space Lm2 −∞, ∞ by the differential expression lyynx P2 x y n−2 x P3 x y n−3 x · · · Pn x y, 1.1 and Lt P2, P3, . . . , Pn ≡ Lt be the differential operator generated in Lm2 0, 1 by the same differential expression and the boundary conditions
We derive an asymptotic formula for the eigenvalues and eigenfunctions of Lt which is uniform with respect to t in Qε n, where
Shkalikov 4, 5 proved that the root functions of the operators generated by an ordinary differential expression with summable coefficients and regular boundary conditions form a Riesz basis with brackets
Summary
Let L P2, P3, . . . , Pn ≡ L be the differential operator generated in the space Lm2 −∞, ∞ by the differential expression lyynx P2 x y n−2 x P3 x y n−3 x · · · Pn x y, 1.1 and Lt P2, P3, . . . , Pn ≡ Lt be the differential operator generated in Lm2 0, 1 by the same differential expression and the boundary conditions. Shkalikov 4, 5 proved that the root functions of the operators generated by an ordinary differential expression with summable coefficients and regular boundary conditions form a Riesz basis with brackets. In general, the eigenvalues are not simple, projections are not uniformly bounded, and Lt has associated function, since the Hill operator with simple potential q x ei2πx has infinitely many spectral singularities see. Using Theorem 1.1, we prove that the spectral expansion of L converges uniformly in every bounded subset of −∞, ∞ if f is absolutely continuous compactly supported function and f ∈ Lm2 −∞, ∞. In this paper, we obtain the spectral expansion for the nonself-adjoint differential operators Lt and L with the periodic matrix coefficients. We do not discuss the results of those papers, since those results have no any relation with the spectral expansion for the nonself-adjoint differential operators Lt and L
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