An eigenfunction expansion for a quadratic pencil of a Schrodinger operator with spectral singularities

In this paper, we consider the operator L generated in L-2(R+) by the diffe rential expression t(y) = -y(n) + [q(x) + 2 lambda p(x) - lambda(2)] y. x is an element of R= [0, infinity), and the boundary condition y(0) = 0, where p and q are complex-valued funct ions and p is continuously differentiable on R+. We derive a two-fold spect ral expansion of L tin the sense of Keldysh, 1951. Soviet Math Dokl. 77, 11 -14 [1971, Russian Math Survey 26, 15-44 (Engl. transl.)]) in terms of the principal functions under the conditions [GRAPHICS] taking into account the spectral singularities. Also we investigate the con vergence of the spectral expansion. (C) 1999 Academic Press.