We consider a general nonselfadjoint second order scalar operator in a multi-dimensional domain with thin spike of finite length. The Dirichlet or Neumann boundary condition is imposed on the boundary. We prove the norm resolvent convergence of such operators to limit operators and estimate the convergence rate in two different operator norms. In the case of the Neumann boundary condition, we show that a one-dimensional operator acting along the spike appears in the limit. Based on the norm resolvent convergence, we establish the convergence of the spectra of perturbed operators to the spectra of the limit ones. In the selfadjoint case, we prove the convergence of spectral projections.
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Translated from Problemy Matematicheskogo Analiza 114, 2022, pp. 15-23.
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Borisov, D.I. Norm Resolvent Convergence of Elliptic Operators in Domains with Thin Spikes. J Math Sci 261, 366–392 (2022). https://doi.org/10.1007/s10958-022-05756-5
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DOI: https://doi.org/10.1007/s10958-022-05756-5