Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let have the form where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spec...

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Bibliographic Details
Published in:Geometric and functional analysis 2023-12, Vol.33 (6), p.1454-1538
Main Authors: Galkowski, Jeffrey, Parnovski, Leonid, Shterenberg, Roman
Format: Article
Language:English
Subjects:
Analysis
Asymptotic methods
Asymptotic properties
Asymptotic series
Differential equations
Mathematics
Mathematics and Statistics
Operators (mathematics)
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Summary:In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let have the form where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, , has a complete asymptotic expansion in powers of ρ . This settles the 1-dimensional case of a conjecture made by the last two authors.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-023-00650-x