Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 057, 27 pages      arXiv:2303.02993      https://doi.org/10.3842/SIGMA.2023.057
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

Modified Green-Hyperbolic Operators

Christopher J. Fewster
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK

Received April 20, 2023, in final form July 29, 2023; Published online August 08, 2023

Abstract
Green-hyperbolic operators – partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators – play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.

Key words: Green-hyperbolic operators; nonlocal equations; Fredholm theory.

pdf (571 kb)   tex (41 kb)  

References

  1. Ammann B., Lauter R., Nistor V., Vasy A., Complex powers and non-compact manifolds, Comm. Partial Differential Equations 29 (2004), 671-705, arXiv:math.OA/0211305.
  2. Bär C., Green-hyperbolic operators on globally hyperbolic spacetimes, Comm. Math. Phys. 333 (2015), 1585-1615, arXiv:1310.0738.
  3. Bär C., Ginoux N., Classical and quantum fields on Lorentzian manifolds, in Global Differential Geometry, Springer Proc. Math., Vol. 17, Springer, Heidelberg, 2012, 359-400, arXiv:1104.1158.
  4. Bär C., Wafo R.T., Initial value problems for wave equations on manifolds, Math. Phys. Anal. Geom. 18 (2015), 7, 29 pages, arXiv:1408.4995.
  5. Bernal A.N., S'anchez M., Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys. 257 (2005), 43-50, arXiv:gr-qc/0401112.
  6. Bernal A.N., S'anchez M., Globally hyperbolic spacetimes can be defined as `causal' instead of `strongly causal', Classical Quantum Gravity 24 (2007), 745-749, arXiv:gr-qc/0611138.
  7. Dappiaggi C., Finster F., Linearized fields for causal variational principles: existence theory and causal structure, Methods Appl. Anal. 27 (2020), 1-56, arXiv:1811.10587.
  8. Davies E.B., Linear operators and their spectra, Cambridge Stud. Adv. Math., Vol. 106, Cambridge University Press, Cambridge, 2007.
  9. Dieudonné J., Schwartz L., La dualité dans les espaces $\mathcal{F}$ et $(\mathcal{LF})$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61-101.
  10. Duistermaat J.J., Hörmander L., Fourier integral operators. II, Acta Math. 128 (1972), 183-269.
  11. Dütsch M., Fredenhagen K., The master Ward identity and generalized Schwinger-Dyson equation in classical field theory, Comm. Math. Phys. 243 (2003), 275-314, arXiv:hep-th/0211242.
  12. Fewster C.J., Jubb I., Ruep M.H., Asymptotic measurement schemes for every observable of a quantum field theory, Ann. Henri Poincaré 24 (2023), 1137-1184, arXiv:2203.09529.
  13. Fewster C.J., Verch R., Measurement of quantum fields on a locally noncommutative spacetime, in preparation.
  14. Fewster C.J., Verch R., A nonlocal generalization of Green hyperbolicity, in preparation.
  15. Geroch R., Spinor structure of space-times in general relativity. I, J. Math. Phys. 9 (1968), 1739-1744.
  16. Gramsch B., Meromorphie in der Theorie der Fredholmoperatoren mit Anwendungen auf elliptische Differentialoperatoren, Math. Ann. 188 (1970), 97-112.
  17. Hawkins E., Rejzner K., The star product in interacting quantum field theory, Lett. Math. Phys. 110 (2020), 1257-1313, arXiv:1612.09157.
  18. Hörmander L., The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics Math., Springer, Berlin, 2003.
  19. Hörmander L., The analysis of linear partial differential operators. III. Pseudo-differential operators, Grundlehren Math. Wiss., Vol. 274, Springer, Berlin, 2007.
  20. Horn R.A., Johnson C.R., Matrix analysis, Cambridge University Press, Cambridge, 1990.
  21. Islam O., Strohmaier A., On microlocalization and the construction of Feynman propagators for normally hyperbolic operators, arXiv:2012.09767.
  22. Kaballo W., Meromorphic generalized inverses of operator functions, Indag. Math. (N.S.) 23 (2012), 970-994.
  23. Lechner G., Verch R., Linear hyperbolic PDEs with noncommutative time, J. Noncommut. Geom. 9 (2015), 999-1040, arXiv:1307.1780.
  24. Lechner G., Waldmann S., Strict deformation quantization of locally convex algebras and modules, J. Geom. Phys. 99 (2016), 111-144, arXiv:1109.5950.
  25. Peetre J., Réctification à l'article ''Une caractérisation abstraite des opérateurs différentiels'', Math. Scand. 8 (1960), 116-120.
  26. Reed M., Simon B., Methods of modern mathematical physics. I. Functional analysis, 2nd ed., Academic Press, New York, 1980.
  27. Seeley R.T., Complex powers of an elliptic operator, in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), American Mathematical Society, Providence, R.I., 1967, 288-307.
  28. Trèves F., Topological vector spaces, distributions and kernels, Dover Publications, Inc., Mineola, NY, 2006.

Previous article  Next article  Contents of Volume 19 (2023)