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

Yu. D. Golovaty; R. O. Hryniv

doi:10.1017/S0308210512000194

Abstract

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For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form

Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

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

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