Spectral regularity with respect to dilations for a class of pseudodifferential operators

2024/11/22 by Horia D. Cornean, Cornean, Horia D., Radu Purice +1
Mathematics · #81Q15 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Primary 81Q10 #Secondary 35S05 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2411.14824

openalex publication_date 2024/11/22 · openalex created_date 2024/12/04 · openalex updated_date 2026/07/28

Abstract

We continue the study of the perturbation problem discussed in \citeCP3 and get rid of the 'slow variation' assumption by considering symbols of the form a(x+δ F(x),ξ) with a a real Hörmander symbol of class S⁰_0,0(ℝ^d×ℝ^d) and F a smooth function with all its derivatives globally bounded, with |δ|≤1. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of δ=0 is still of the order √(|δ|), the distance between their spectral edges behaves like |δ|^ν with ν∈[1/2,1) depending on the rate of decay of the second derivatives of F at infinity.

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