COMMUNICATIONS Resolvent Convergence of Sturm–Liouville Operators with Singular Potentials

Suppose that the formal differential expression l(y) = −y′′(t) + q′(t)y(t), q ( · ) ∈ L2([a, b],C) =: L2, (1.1) is given on a compact interval [a, b]. In rigorous terms, it can be defined as quasidifferential, l(y):= D[2]y, by introducing the quasiderivatives [1]: D[0]y = y, D[1]y = y ′ − qy, D[2]y = −(D[1]y) ′ − qD[1]y − q2y. Consider the family of quasidifferential expressions lε ( · ) of the form (1.1) with potentials qε ( · ) ∈ L2, ε ∈ [0, ε0]. In the Hilbert space L2 with norm ‖ · ‖2, with each of these expressions we can associate a closed densely given quasidifferential operator Lεy: = lε(y), Dom(Lε): = {y ∈ L2: ∃D[2]ε y ∈ L2, α(ε)Ya(ε) + β(ε)Yb(ε) = 0}, where the matrices α(ε), β(ε) ∈ C2×2 and the vectors Ya(ε): = {y(a),D[1]ε y(a)}, Yb(ε): = {y(b),D[1]ε y(b)} ∈ C2. Let us recall that the family of operators Lε converges to L0 in the sense of norm resolvent convergence, Lε R− → L0, if there exists a number μ ∈ C belonging to the resolvent sets ρ(L0) and ρ(Lε) (for all sufficiently small ε) and ‖(Lε − μ)−1 − (L0 − μ)−1 ‖ → 0, ε → +0. This definition is independent of the choice of the point μ ∈ ρ(L0) [2]. For the case in which the matrices α(ε), β(ε) are independent of ε, the following important theorem was established in [1]. Theorem 1. Suppose that ‖qε − q0‖2 → 0 as ε → +0 and the resolvent set of the operator L0 is not empty. Then L