Schroedinger operators with (\alpha\delta'+\beta \delta)-like potentials: norm resolvent convergence and solvable models

Abstract

For real functions \Phi and \Psi that are integrable and compactly supported, we prove the norm resolvent convergence, as \epsilon\ goes to 0, of a family S(\epsilon) of one-dimensional Schroedinger operators on the line of the form S(\epsilon)= -D^2 + \alpha \epsilon^{-2} \Phi(x/\epsilon) + \beta \epsilon^{-1} \Psi(x/\epsilon). 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 (\Phi,\Psi). We show that it is impossible to assign just one self-adjoint operator to the pseudo-Hamiltonian -D^2 + \alpha \delta'(x) + \beta \delta(x), which is a symbolic notation only for a wide variety of quantum systems with quite different properties.