Author(s): Sobhy El-Sayed Ibrahim

Abstract: In this paper, we consider the problem of a general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ⁺ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation (τ-λw)u=wf are in the weighted Hilbert space L_w² (a,b) provided that all solutions of the equations (τ-λw)u=0 and its adjoint (τ⁺-¯λ w)v=0 are in L_w² (a,b) (the limit circle case). Also, under suitable conditions on the growth of the coefficients, the resolvent operator is proved to be Hilbert-Schmidt integral operator, and a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.

Pages: 115-127 | Views: 433 | Downloads: 36

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