Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

2023/02/28 by Cornean, Horia D., Purice, Radu
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2303.00112

Abstract

Let a(x,ξ) be a real Hörmander symbol of the type S_0,0⁰(ℝ^d× ℝ^d), let F be a smooth function with all its derivatives globally bounded, and let K_δ be the self-adjoint Weyl quantization of the perturbed symbols a(x+F(δ x),ξ), where |δ|≤ 1. First, we prove that the Hausdorff distance between the spectra of K_δ and K₀ is bounded by √(|δ|), and we give examples where spectral gaps of this magnitude can open when δ≠ 0. Second, we show that the distance between the spectral edges of K_δ and K₀ (and also the edges of the inner spectral gaps, as long as they remain open at δ=0) are of order |δ|, and give a precise dependence on the width of the spectral gaps.

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