Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

Source abstract

For a real-valued function V of the Faddeev–Marchenko class, 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 Under certain conditions, the functions ε −2 V ( x / ε ) converge in the sense of distributions as ε → 0 to δ ′ ( x ), and then the limit S 0 of S ε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ ′.