Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = − u ″ + q ( x ) u on the interval G = (0, 1) with complex-valued potential q ( x ) locally integrable on G and satisfying the inequalities ∫ x 1 x 2 ζ | ( q ( ζ ) ) | d ζ ≤ l n ( x 1 / x 2 ) and ∫ x 1 x 2 ζ | ( q ( 1 − ζ ) ) | d ζ ≤ γ l n ( x 1 / x 2 ) with some constant γ for all sufficiently small 0 < x 1 < x 2 , we estimate the norms of root functions in the Lebesgue spaces L p ( G ), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.