The Sturm-Liouville Problem with a Potential Linear in Spectral Parameter

Consider the boundary problem $$ - y{\text{}} + [\lambda ^2 + U(x) + 2\lambda Q(x)]y = 0, $$ (1) $$y(\lambda ,0) = 0$$ (2) where y ∈ L2(0, ∞) is an unknown function, λ is the spectral parameter. The following conditions are assumed to be satisfied throughout this paper: 1) U(x) is real a function continuous on the semiaxis (0, ∞) and \(\int\limits_0^\infty {} U(X)|xdx < \infty \) 2) Q(x) is real a function continuously differentiable on \([0,\infty ),\int\limits_0^\infty {|Q(x)} |xdx < \infty ,\int\limits_0^\infty {|{Q^,}} (x)|xdx < \infty \) 3) Q(x)≥ O.