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The inverse backscattering for Schrödinger operators for potentials with noncompact support
Author(s) -
Huang Hua,
Huang Shanlin,
Zheng Quan,
Duan Zhiwen
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5586
Subject(s) - mathematics , operator (biology) , schrödinger's cat , infinity , inverse , diffeomorphism , banach space , inverse problem , mathematical physics , mathematical analysis , order (exchange) , pure mathematics , geometry , biochemistry , chemistry , finance , repressor , transcription factor , economics , gene
We consider the inverse backscattering problem for the Schrödinger operator H = −Δ + V on R n , n ≥ 3, as well as the higher‐order Schrödinger operator ( − Δ) m + V , m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that V ∈ C 0 ∞ . In this paper, we replace theC 0 ∞assumption on V with certain decay assumption at infinity.