1D Schrodinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real L-infinity(R)-functions Phi and Psi of compact support, we prove the norm resolvent convergence, as epsilon and nu tend to 0, of a family S-epsilon nu of one-dimensional Schrodinger operators on the line of the form S-epsilon nu = -d(2)/dx(2) + alpha/epsilon(2) Phi (x/epsilon) + beta/nu Psi (x/nu), provided the ratio nu/epsilon has a finite or infinite limit. The limit operator S-0 depends on the shape of Phi and Psi as well as on the limit of ratio nu/epsilon. If the potential alpha Phi possesses a zero-energy resonance, then S-0 describes a non trivial point interaction at the origin. Otherwise S-0 is the direct sum of the Dirichlet half-line Schrodinger operators.