An Eigenfunction Expansion for a Quadratic pencil of a Schrodinger Operator with Spectral Singularities

摘要

In this paper, we consider the operator L generated in L~2(R_+) by the differential expression L(y) = y~n + [q(x) + 2#lambda#p(x) - #lambda#~2]y, the point x belong to (is member of) the set R_+ = [0,infinity), and the boundary condition y(0) = 0, where p and q are complex-valued functions and p is continuously differentiable on R_+. We derive a two-fold spectral expansion of L (in the sense of Keldysh. 1951. Soriet Math. Dokl. 77, 11-14 [1971, Russian Math. Survey 26, 15-44 (Engl. transl. )]) in terms of the principal functions under the conditions lim (x->infinity) p(x) = 0, sup (the point x belong to (is member of) the set R_perpendicular {e~(#epsilon#x)[|q(x)| + |p'(x)|]} 0, taking into account the spectral singularities. Also we investigate the convergence of the spectral expansion.