Point interaction in dimension two and three as models of small scatterers

Pavel Exner P eba

TBD mathscidoc:1910.43411

arXiv preprint cond-mat/9607016
Point interactions described by formal shaped potentials were used in quantum mechanics from the early thirties. It lasted several decades, however, until a proper way to handle the corresponding Schrdinger operators was found. Following the observation of Berezin and Faddeev [4], the problem was studied systematically in the eighties. The results are summarized in the monograph [1], references to some recent work are given, eg, in [2, 8]. In its present form the pointinteraction method represents a versatile tool for constructing solvable models whose power has not been yet, to our opinion, appreciated fully in the physical community. Within the standard quantum mechanical formalism point interactions can be constructed as long as the configuration space dimension does not exceed three, and they can be interpreted as limits of suitable families of squeezed potentials. In the onedimensional case the limiting argument admits a straightforward interpretation, because the pointinteraction coupling constant is just the integral of the approximating potential: a slow particle on the line with a wellspread wavefunction sees only the average value of a localized potential. In distinction to that the approximation by scaled potentials in dimension two and three requires existence of a zeroenergy resonance and a particular couplingconstant renormalization; a detailed discussion of the related mathematical problems can be found in [1, 9]. This gives these point interactions a flavour of something exceptional, for instance, one is led to believe that they cannot be used to model welllocalized repulsive potentials. That would have consequences
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  title={Point interaction in dimension two and three as models of small scatterers},
  author={Pavel Exner, and P eba},
  booktitle={arXiv preprint cond-mat/9607016},
Pavel Exner, and P eba. Point interaction in dimension two and three as models of small scatterers. In arXiv preprint cond-mat/9607016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020144102652218940.
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