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Letters in Mathematical Physics

, Volume 109, Issue 12, pp 2625–2648 | Cite as

A note on the Schrödinger operator with a long-range potential

  • D. R. YafaevEmail author
Article

Abstract

Our goal is to develop spectral and scattering theories for the one-dimensional Schrödinger operator with a long-range potential q(x), \(x\ge 0\). Traditionally, this problem is studied with a help of the Green–Liouville approximation. This requires conditions on the first two derivatives \(q' (x)\) and \(q'' (x)\). We suggest a new Ansatz that allows us to develop a consistent theory under the only assumption \(q' \in L^1\).

Keywords

Schrödinger equation Dimension one Modified Green–Liouville Ansatz Limiting absorption principle Eigenfunction expansion 

Mathematics Subject Classification

34E20 34L10 34L15 47A40 81U05 

Notes

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CNRS, IRMAR-UMR 6625Univ RennesRennesFrance
  2. 2.SPGUSaint PetersburgRussia

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