Absence of reflection as a function of the coupling constant

Rowan Killip, Robert Sims

Research output: Contribution to journalArticle

1 Scopus citations

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) = u0(x) for all x<0, we prove that the zeros of b(λ) := W[u0, uλ], the Wronskian of u0 and uλ, 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 languageEnglish (US)
Article number062102
JournalJournal of Mathematical Physics
Volume47
Issue number6
DOIs
StatePublished - Jul 11 2006
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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