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

Abstract

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 \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, 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.