Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

2020/11/18 by Jeffrey Galkowski, Galkowski, Jeffrey
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2011.09245

openalex publication_date 2020/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let P:L²(ℝ)→ L²(ℝ) have the form P:=-\tfracd²dx²+W, where W is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, \mathbb1_(-∞,λ²](P) has a full asymptotic expansion in powers of λ. In particular, our class of potentials W is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.

Related

• Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case
• Spectral shift function for slowly varying perturbation of periodic\n Schroedinger operators
• Formulas of Szegő type for the periodic Schrödinger operator
• A representation formula related to Schrodinger operators
• Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators