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

Research output

Abstract

Our goal is to develop spectral and scattering theories for the one-dimensional Schrödinger operator with a long-range potential q(x), x≥ 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 ∈ L 1.

Original languageEnglish
Pages (from-to)2625-2648
JournalLetters in Mathematical Physics
Volume109
Issue number12
Early online date15 Jul 2019
DOIs
Publication statusPublished - Dec 2019

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spectral theory
Scattering Theory
Spectral Theory
Derivative
operators
Approximation
Operator
approximation
scattering
Range of data

Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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title = "A note on the Schr{\"o}dinger operator with a long-range potential",
abstract = "Our goal is to develop spectral and scattering theories for the one-dimensional Schr{\"o}dinger operator with a long-range potential q(x), x≥ 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 ′∈ L 1.",
keywords = "Dimension one, Eigenfunction expansion, Limiting absorption principle, Modified Green–Liouville Ansatz, Schr{\"o}dinger equation",
author = "D.R. Yafaev",
note = "Yafaev, D.R. A note on the Schr{\"o}dinger operator with a long-range potential. Lett Math Phys 109, 2625–2648 (2019) doi:10.1007/s11005-019-01200-z",
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N2 - Our goal is to develop spectral and scattering theories for the one-dimensional Schrödinger operator with a long-range potential q(x), x≥ 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 ′∈ L 1.

AB - Our goal is to develop spectral and scattering theories for the one-dimensional Schrödinger operator with a long-range potential q(x), x≥ 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 ′∈ L 1.

KW - Dimension one

KW - Eigenfunction expansion

KW - Limiting absorption principle

KW - Modified Green–Liouville Ansatz

KW - Schrödinger equation

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