Schrödinger operators with (αδ’+β δ)-like potentials: norm resolvent convergence and solvable models - Golovaty, Yu. (Methods of Functional Analysis and Topology, 2012)
For real functions Φ and Ψ that are integrable and compactly supported, we prove the norm resolvent convergence, as ε→0, of a family S_ε of one-dimensional Schrödinger operators on the line of the form S_ε= -d^2d x^2+αε^-2Φ(ε^-1x)+βε^-1Ψ(ε^-1x). The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator S_0 exists and strongly depends on the pair (Φ,Ψ). A class of nontrivial point interactions which are formally related the pseudo-Hamiltonian -d^2dx^2+αδ’(x)+βδ(x) is singled out. The limit behavior, as ε→ 0, of the scattering data for such potentials is also described.