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Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

Author

Listed:
  • 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
    Southern Mathematical Institute, Vladikavkaz Scientific Center of RAS, Vladikavkaz 362025, Russia)

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.

Suggested Citation

  • Denis Ivanovich Borisov & Dmitry Mikhailovich Polyakov, 2023. "Resolvent Convergence for Differential–Difference Operators with Small Variable Translations," Mathematics, MDPI, vol. 11(20), pages 1-33, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4260-:d:1258316
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    References listed on IDEAS

    as
    1. A. L. Skubachevskii, 2018. "Elliptic differential‐difference operators with degeneration and the Kato square root problem," Mathematische Nachrichten, Wiley Blackwell, vol. 291(17-18), pages 2660-2692, December.
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