### Abstract

This paper studies the scattering matrix S (E ; ?) of the problem- ?^{2} ?^{?} (x) + V (x) ? (x) = E ? (x) for positive potentials V ? C^{8} (R) with inverse square behavior as x ? ± 8. It is shown that each entry takes the form S_{i j} (E ; ?) = S_{i j}^{(0)} (E ; ?) (1 + ? s_{i j} (E ; ?)) where S_{i j}^{(0)} (E ; ?) is the WKB approximation relative to the modified potentialV (x) + frac(?^{2}, 4) <x >^{-2} and the correction terms s_{i j} satisfy | ?_{E}^{k} s_{i j} (E ; ?) | = C_{k} E^{- k} for all k = 0 and uniformly in (E, ?) ? (0, E_{0}) × (0, ?_{0}) where E_{0}, ?_{0} are small constants. This asymptotic behavior is not universal: if - ?^{2} ?_{x}^{2} + V has a zero energy resonance, then S (E ; ?) exhibits different asymptotic behavior as E ? 0. The resonant case is excluded here due to V > 0. © 2008 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 2321-2362 |

Number of pages | 42 |

Journal | Journal of Functional Analysis |

Volume | 255 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Nov 2008 |

### Keywords

- Inverse square potential
- Modified WKB
- Scattering matrix
- Schrödinger operators
- Zero energy scattering

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## Cite this

*Journal of Functional Analysis*,

*255*(9), 2321-2362. https://doi.org/10.1016/j.jfa.2008.07.015