QUADRATIC PENCIL OF SCHRODINGER OPERATORS WITH SPECTRAL SINGULARITIES - DISCRETE SPECTRUM AND PRINCIPAL FUNCTIONS

摘要

In this article we investigated the spectrum of the quadratic pencil of Schrodinger operators L(lambda) generated in L-2(R+) by the equation -y '' + [V(x) + 2 lambda U(x) - lambda(2)]y = 0, x is an element of R+ = [0,infinity) and the boundary condition y(0) = 0, where U,V are complex valued functions and U is absolutely continuous in each finite subinterval of R+. Discussing the spectrum, we proved that L(lambda) has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions sup {exp(is an element of root x)[V(X) + U'(x)]} < infinity, is an element of > 0 0 less than or equal to x < infinity and lim U(x) = 0 x --> infinity hold. It is shown that the principal functions corresponding to eigenvalues of L(lambda) are in the space L-2(R+), and the principal functions corresponding to spectral singularities are in another Hilbert space, which contains L-2(R+). Some results about the spectrum of L(lambda) have also been applied to radial Klein-Gordon and one-dimensional Schrodinger equations. (C) 1997 Academic Press. [References: 20]