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

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

2011-08-26

arxiv.org/abs/1108.5345v2

Mathematics

Spectral Theory

30 pages, 2 figure; submitted to Proceedings of the Royal Society of Edinburgh

Scientific paper

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:= -D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_\epsilon might be considered as a "physically motivated" interpretation of the one-dimensional Schr\"odinger operator with potential \delta'.

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