Index theory for Fourier integral operators
Final Report Abstract
Darstellung der wichtigsten wissenschaftlichen Fortschritte/Presentation of Most Relevant Scientific Results: In a unified approach to various problems in index theory, we consider operators of the form (1) D = cI + Dg Φg dg : L2 (M ) → L2 (M ), G acting on L2 -functions on a closed manifold M . Here, c ∈ C, G is a finitedimensional Lie group, g → Dg is a smooth, compactly supported family of pseudodifferential operators of order zero, and g → Φg is a unitary representation of G by quantized canonical transformations. In particular, G might be a discrete group; then we obtain finite sums of operators (2) D= Dg Φg : L2 (M ) → L2 (M ). g∈G To an operator D as in (1) we associated the symbol ∗ σ(D) = c + {σ(Dg )}g∈G ∈ (C(SG M ) ⋊ G)+ , ∗ in the unitization of the maximal crossed product of C(SG M ) and G. In this context, SG ∗ M is the sphere in the so-called transverse cotangent space ∗ ∗ TG M . In the case of a discrete group, SG M = S ∗ M . We call D elliptic, if σ(D) is invertible. In this case D turns out to be a Fredholm operator. In order to obtain an index formula, we pursued an approach via deformation theory, focusing on the case of a discrete group G with operators as in (2). We assumed D to be elliptic and the symbol σ(D) to have an inverse in the algebraic crossed product C ∞ (S ∗ M ) ⋊alg G. For a finite order element g of G we introduced both an analytic and an algebraic index, localized at the conjugacy class g of g, and showed that both coincide. The definition of the algebraic index relies on the construction of an adapted semiclassical calculus and a trace formula establishing an asymptotic expansion in terms of powers of the semiclassical variable h for operators of the form oph (a)Φh , where oph (a) is the quantization of a semiclassical pseudodifferential symbol of sufficiently low order and Φh is a semiclassical quantized canonical transformation. If G is a finite extension of Zn for some n ∈ N we obtain the following formula for the index of D in terms of the localized algebraic indices indalg (D), g g ∈ G: ind D = indalg (D) g g ⊆Tor Gwith the sum over all torsion conjugacy classes in G. In related work, Nazaikinskii, Savin and Sipailo considered nonlocal problems of Sobolev type on a closed manifold with conditions on a submanifold involving the spherical means operator. This leads to the analysis of operators of the form (2). Using the above results, they established the Fredholm property and computed the index. The project suffered from the unexpected death of Boris Sternin, a great mathematician and the driving force behind these investigations. He died in the spring of 2017, and we missed his expertise on many occasions.
Publications
- Elliptic operators associated with groups of quantum canonical transformations. (Russian) Uspekhi Mat. Nauk 73, no. 3(441):179-180 (2018), translation: Russian Math. Surveys 73(3):546-548 (2018)
A. Yu. Savin, E. Schrohe, B. Yu. Sternin
(See online at https://doi.org/10.1070/RM9830) - On traces of Fourier integral operators on submanifolds (Russian) Mat. Zametki 104 (2018), no. 4, 588–603, translation: Math. Notes 104 (2018), no. 3-4, 559–571
P. A. Sipailo
(See online at https://doi.org/10.1134/S0001434618090225) - Elliptic operators associated with groups of quantized canonical transformations. Bull. Sci. Math.155:141-167 (2019)
A. Yu. Savin, E. Schrohe, B. Yu. Sternin
(See online at https://doi.org/10.1016/j.bulsci.2019.01.010) - Operator algebras associated with quantized canonical transformations. RIMS Kôkyûroku volume 2137
A. Savin, E. Schrohe
- Traces of quantized canonical transformations on submanifolds and their applications to Sobolev problems with nonlocal conditions. Russ. J. Math. Phys. 26 (2019), no. 1, 135–138
P. A. Sipailo
(See online at https://doi.org/10.1134/S106192081901014X) - A class of Fredholm boundary value problems for the wave equation with conditions on the entire boundary. In Differential equations on manifolds and mathematical physics, Trends in mathematics. Birkhäuser, 2020
A. Boltachev and A. Savin
- Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations. J. Funct. Anal. 278:108400 (2020)
A. Savin, E. Schrohe
(See online at https://doi.org/10.1016/j.jfa.2019.108400)
