Such operators also arose in connection with the approximation of smooth planar quantum waveguides by quantum graph a similar resonance phenomenon was obtained. The zero-energy resonances have a profound effect on the limiting behavior of the Schrödinger operators with δ′-potentials. , Golovaty and Hryniv, and Man'ko these resonances deal with the existence of zero-energy resonances and the half-bound states for singular localized potentials. Recently a class of the Schrödinger operators with piece-wise constant δ′-potentials were studied by Zolotaryuk the resonances in the transmission probability for the scattering problem were established. Therefore, the pseudo-Hamiltonians (1.2) can be regarded as a symbolic notation only for a wide variety of quantum systems with quite different properties depending on the shape of the short-range potentials. So it is not surprising that different regularizations of the distributions in (1.2) lead to different self-adjoint operators in the limit. However, both the heuristic operators have generally no mathematical meaning. Thus we may formally regard the δ′ potential as rank-two perturbation δ′( x) y = 〈 δ( x), y〉 δ′( x) + 〈 δ′( x), y〉 δ( x). We note that δ′( x) y = y(0) δ′( x) − y′(0)δ( x) for continuously differentiable functions y at the origin.
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