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Some New and Old Asymptotic Representations of the Jost Solution and the Weyl m ‐Function For Schrödinger Operators on the Line
Author(s) -
Rybkin Alexei
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008645
Subject(s) - mathematics , representation (politics) , operator (biology) , line (geometry) , term (time) , function (biology) , series (stratigraphy) , real line , pure mathematics , mathematical physics , asymptotic expansion , mathematical analysis , quantum mechanics , physics , geometry , paleontology , biochemistry , chemistry , repressor , evolutionary biology , politics , biology , political science , transcription factor , law , gene
For the general one‐dimensional Schrödinger operator − d 2 / dx 2 + q ( x ) with real q ∈ L 1 (R), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m ‐function for locally summable q . This representation is then applied to smooth potentials q to obtain Weyl m ‐function power asymptotics. The condition q ( N ) ∈ L 1 ( x 0 , x 0 + δ), for N ∈ N 0 , allows one to derive the ( N + 1) term for almost all x ∈ [ x 0 , x 0 + δ), thereby refining a relevant result by Danielyan, Levitan and Simon. 2000 Mathematics Subject Classification 34E05, 34L40 (primary), 34B20, 34L25 (secondary).