A New Class of Point Interactions in One Dimension

We present a class of self-adjoint extensions of the symmetric operator −Δ|C∞0(R1\{0}) which correspond formally to perturbations of the Laplacian by pseudopotentials involving δ2. These operators, which provide new examples of generalized point interactions in the sense of Šeba, are defined by the boundary conditions ƒ(0+) = e−zƒ(0−), rƒ(0+) + ƒ′(0+) = ez[rƒ(0−) + ƒ′(0−)], for z ∈ C, r ∈ R. We calculate their spectra, resolvents, and scattering matrices, and show that they can be realized as limits of Schrödinger operators with local short-range potentials.