On two methods of determining η-invariants of elliptic boundaryvalue problems

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Abstract

For a class of boundary-value problems with a parameter that are elliptic in the sense of Agranovich-Vishik, we establish the equality of the η-invariant defined in terms of the Melrose regularization and the spectral η-invariant of the Atiyah-Patodi-Singer type defined using the analytic continuation of the spectral η-function of the operator.

About the authors

K. N. Zhuikov

RUDN University

Author for correspondence.
Email: zhuykovcon@gmail.com
Moscow, Russia

A. Yu. Savin

RUDN University

Email: a.yu.savin@gmail.com
Moscow, Russia

References

  1. Агранович М.С., Вишик М.И. Эллиптические задачи с параметром и параболические задачи общего вида// Усп. мат. наук.- 1964.- 19, № 3.-С. 53-161.
  2. Бейтмен Г., Эрдейи А. Высшие трансцендентные функции, Т. 1.- М.: Наука, 1973.
  3. Жуйков К.Н., Савин А.Ю. Эта-инвариант эллиптических краевых задач с параметром// Соврем. мат. Фундам. направл.-2023.-69, № 4.- С. 599-620.
  4. Кондратьев В.А. Краевые задачи для эллиптических уравнений в областях с коническими и угловыми точками// Тр. Моск. мат. об-ва.-1967.- 16.-С. 209-292.
  5. Atiyah M., Patodi V., Singer I. Spectral asymmetry and Riemannian geometry. I// Math. Proc. Cambridge Philos. Soc.- 1975.- 77.- С. 43-69.
  6. Atiyah M., Patodi V., Singer I. Spectral asymmetry and Riemannian geometry. II// Math. Proc. Cambridge Philos. Soc.- 1976.- 78.- С. 405-432.
  7. Atiyah M., Patodi V., Singer I. Spectral asymmetry and Riemannian geometry. III// Math. Proc. Cambridge Philos. Soc.- 1976.- 79.-С. 71-99.
  8. Fedosov B., Schulze B.-W., Tarkhanov N. The index of elliptic operators on manifolds with conical points// Selecta Math. (N.S.). -1999.- 5, № 4.-С. 467-506.
  9. Fedosov B., Schulze B.-W., Tarkhanov N. A general index formula on toric manifolds with conical points// В сб.: «Approaches to singular analysis».-Basel: Birkh¨auser, 2001.- С. 234-256.
  10. Gilkey P.B., Smith L. The eta invariant for a class of elliptic boundary value problems// Commun. Pure Appl. Math. -1983.- 36.- С. 85-132.
  11. Gilkey P.B., Smith L. The twisted index problem for manifolds with boundary// J. Differ. Geom.- 1983.- 18, № 3.- С. 393-444.
  12. Lesch M. Differential Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods.- Stuttgart-Leipzig: B.G. Teubner Verlag, 1997.
  13. Lesch M., Pflaum M. Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant// Trans. Am. Math. Soc. -2000.- 352, № 11.- С. 4911-4936.
  14. Lidskii V.B. Non-selfadjoint operators with a trace// Dokl. Akad. Nauk SSSR. -1959.- 125.-С. 485- 487.
  15. Melrose R. The eta invariant and families of pseudodifferential operators// Math. Research Lett. - 1995.- 2, № 5.-С. 541-561.

Copyright (c) 2024 Zhuikov K.N., Savin A.Y.

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