### Abstract

We consider solutions of the one-dimensional equation - u″+(Q+λV)u=0 where Q:ℝ→ℝ is locally integrable, V:ℝ→ℝ is integrable with supp(V) ⊂[0, 1], and λ ∈ ℝ is a coupling constant. Given a family of solutions {u _{λ}}_{λ∈ℝ} which satisfy u _{λ}(x) = u_{0}(x) for all x<0, we prove that the zeros of b(λ) := W[u_{0}, u_{λ}], the Wronskian of u_{0} and u_{λ}, form a discrete set unless V≡0. Setting Q(x) :=-E, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment -u″+λVu=Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V≡0.

Original language | English (US) |
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Article number | 062102 |

Journal | Journal of Mathematical Physics |

Volume | 47 |

Issue number | 6 |

DOIs | |

State | Published - Jul 11 2006 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Killip, R., & Sims, R. (2006). Absence of reflection as a function of the coupling constant.

*Journal of Mathematical Physics*,*47*(6), [062102]. https://doi.org/10.1063/1.2206691