Wildly perturbed manifolds: norm resolvent and spectral convergence

The publication of the important work Rauch and Taylor [RT75] started a hole branch research on wild perturbations Laplace-Beltrami operator. Here, we extend certain results show norm convergence resolvent. We consider (not necessarily compact) manifold with many small balls removed, number can increase as radius is shrinking, also be infinite. If distance shrinks less fast than radius, then that Neumann Laplacian converges to unperturbed Laplacian, i.e., obstacles vanish. In Dirichlet case, two cases here: if are too sparse, limit operator again one, while concentrate at region (they become solid there), complement region. Norm resolvent in case ho-mogenisation treated elsewhere, see [KP18] references therein. Our based result for operators acting varying Hilbert spaces described book [P12] by second author.