Eta-invariant for parameter-dependent families with periodic coefficients

K. N. Zhuikov   ; A. Yu. Savin

Ufa mathematical journal, Tome 14 (2022) no. 2, pp. 35-55

Résumé

On a closed smooth manifold, we consider operator families being linear combinations of parameter-dependent pseudodifferential operators with periodic coefficients. Such families arise in studying nonlocal elliptic problems on manifolds with isolated singularities and/or with cylindrical ends. The aim of the work is to construct the $\eta$-invariant for invertible families and to study its properties. We follow Melrose's approach who treated the $\eta$-invariant as a generalization of the winding number being equal to the integral the trace of the logarithmic derivative of the family. At the same time, the Melrose $\eta$-invariant is equal to the regularized integral of the regularized trace of the logarithmic derivative of the family. In our situation, for the trace regularization, we employ the operator of difference differentiating instead of the usual differentation used by Melrose. The main technical result is the fact that the operator of difference differentiation is an isomorphism between the spaces of functions with conormal asymptotics at infinity and this allows us to determine the regularized trace. Since the obtained regularized trace can increase at infinity, we also introduce a regularization for the integral. Our integral regularization involves an averaging operation. Then we establish the main properties of the $\eta$-invariant. Namely, the $\eta$-invariant in the sense of this work satisfies the logarithmic property and is a generalization of Melrose's $\eta$-invariant, that is, it coincides with it for usual parameter-dependent pseudodifferential operators. Finally, we provide a formula for the variation of the $\eta$-invariant under a variation of the family.

Détail

Keywords: elliptic operator, parameter-dependent operator, difference differentiation.
Mots-clés : $\eta$-invariant

Bibliographie

[1] M.S. Agranovich, M.I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type”, Russ. Math. Surv., 19:3 (1964), 53–157

[2] A.Yu. Savin, B.Yu. Sternin, “Elliptic ${G}$-operators on manifolds with isolated singularities”, J. Math. Sci., 233:6 (2018), 930–948

[3] M.A. Shubin, Pseudodifferential operators and spectral theory, Springer, Berlin, 2001

[4] M. Atiyah, V. Patodi, I. Singer, “Spectral asymmetry and Riemannian geometry I”, Math. Proc. Cambridge Philos. Soc., 77 (1975), 43–69

[5] J.-M. Bismut, J. Cheeger, “$\eta$-invariants and their adiabatic limits”, J. of Amer. Math. Soc., 2:1 (1989), 33–70

[6] J. Cheeger, “$\eta$-invariants, the adiabatic approximation and conical singularities”, J. Diff. Geometry, 26:1 (1987), 175–221

[7] Yu. Egorov, B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Birkhäuser, Boston–Basel–Berlin, 1997

[8] B.V. Fedosov, B.-W. Schulze, N. Tarkhanov, “The index of higher order operators on singular surfaces”, Pacific J. of Math., 191:1 (1999), 25–48

[9] M. Lesch, H. Moscovici, M.J. Pflaum, Connes-Chern character for manifolds with boundary and eta cochains, Mem. Amer. Math. Soc., 220, no. 1036, 2012, viii+92 pp.

[10] M. Lesch, M. Pflaum, “Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant”, Trans. Amer. Math. Soc., 352:11 (2000), 4911–4936

[11] R. Melrose, “The eta invariant and families of pseudodifferential operators”, Math. Research Letters, 2:5 (1995), 541–561

[12] R. Melrose, F. Rochon, “Eta forms and the odd pseudodifferential families index”, Surveys in differential geometry, v. XV, Surv. Differ. Geom., 15, Perspectives in mathematics and physics, Int. Press, Somerville, MA, 2011, 279–322

[13] W. Müller, “Eta-invariant (some recent developments)”, Sem. Bourbaki, Asterisque, 227, 1994, 335–364

[14] V. Nazaikinskii, A. Savin, B.-W. Schulze, B. Sternin, Elliptic Theory on Singular Manifolds, CRC-Press, Boca Raton, 2005

[15] V. Nistor., “An index theorem for gauge-invariant families: The case of solvable groups”, Acta Math. Hungarica, 99:2 (2003), 155–183

[16] A.L. Skubachevskii, Elliptic functional differential equations and applications, Birkhäuser, Basel–Boston–Berlin, 1997

[17] A.L. Skubachevskii, “Nonlocal elliptic problems in infinite cylinder and applications”, Discrete Contin. Dyn. Syst. Ser. S, 9:3 (2016), 847–868

[18] E. Witten, “Quantum field theory and the Jones polynomial”, Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys., 9, World Sci. Publ., Teaneck, NJ, 1989, 239–329