Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

Denis Ivanovich Borisov () and Dmitry Mikhailovich Polyakov
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Denis Ivanovich Borisov: Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa 450008, Russia
Dmitry Mikhailovich Polyakov: Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa 450008, Russia

Mathematics, 2023, vol. 11, issue 20, 1-33

Abstract: We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra.

Keywords: differential–difference operator; higher-order operator; small translation; norm resolvent convergence; operator estimate; spectrum; pseudospectrum (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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