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

We continue the study of the perturbation problem discussed in H. D. Cornean and R. Purice (2023) 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 S⁰0,0(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.