### 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 language | English |
---|---|

Pages (from-to) | 2625-2648 |

Journal | Letters in Mathematical Physics |

Volume | 109 |

Issue number | 12 |

Early online date | 15 Jul 2019 |

DOIs | |

Publication status | Published - Dec 2019 |

### Fingerprint

### Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Letters in Mathematical Physics*, vol. 109, no. 12, pp. 2625-2648. https://doi.org/10.1007/s11005-019-01200-z

**A note on the Schrödinger operator with a long-range potential.** / Yafaev, D.R.

Research output

TY - JOUR

T1 - A note on the Schrödinger operator with a long-range potential

AU - Yafaev, D.R.

N1 - Yafaev, D.R. A note on the Schrödinger operator with a long-range potential. Lett Math Phys 109, 2625–2648 (2019) doi:10.1007/s11005-019-01200-z

PY - 2019/12

Y1 - 2019/12

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

UR - http://www.scopus.com/inward/record.url?scp=85068992156&partnerID=8YFLogxK

U2 - 10.1007/s11005-019-01200-z

DO - 10.1007/s11005-019-01200-z

M3 - Article

VL - 109

SP - 2625

EP - 2648

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 12

ER -