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

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_ı be the self-adjoint Weyl quantization of the perturbed symbols a.x C F.ıx/; ξ/, where jıj ≤ 1. First, we prove that the Hausdorff distance between the spectra of K_ı and K₀ is bounded by pjıj, and we give examples where spectral gaps of this magnitude can open when ı ¤ 0. Second, we show that the distance between the spectral edges of K_ı and K₀ (and also the edges of the inner spectral gaps, as long as they remain open at ı D 0) are of order jıj, and give a precise dependence on the width of the spectral gaps.