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Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary

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Abstract

We consider the wave operator on static, Lorentzian manifolds with timelike boundary, and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus, we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated elliptic operator on a Riemannian manifold with boundary (M, g). The latter is diffeomorphic to any constant time hypersurface of the underlying background. In turn, assuming that (M, g) is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace operators from the domain of the adjoint of the elliptic operator onto an auxiliary Hilbert space \({\mathsf {h}}\), which is the third datum of the triple. Self-adjoint extensions of the underlying elliptic operator are in one-to-one correspondence with self-adjoint operators \(\Theta \) on \({\mathsf {h}}\). On the one hand, we show that, for a natural choice of boundary triple, each \(\Theta \) can be interpreted as the assignment of a boundary condition for the original wave operator. On the other hand, we prove that, for each such \(\Theta \), there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated with the wave operator on a globally hyperbolic spacetime.

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Notes

  1. Recall that, if we consider the restriction map \(\text {res}:{\widehat{M}}\rightarrow M\), then \(\text {res}\circ \iota =\iota \). Hence, \(\iota ^*{\widehat{g}}=\iota ^*g\).

  2. The orthogonal decomposition \(\oplus _S\) refers to the scalar product on \(D(S^*)\) defined by \((f\,|\,f')_S:=(f\,|\,f')\,+(S^*f\,|\,S^*f').\)

  3. In this paper, \(V\dotplus W\) denotes the direct sum between the subspaces \(V,W\subseteq {\mathsf {H}}\) (\(V+W={\mathsf {H}}\) and \(V\cap W=\{0\}\)), while \(V\oplus W\) denotes the orthogonal direct sum (\(V+W={\mathsf {H}}\) and \(V\perp W\)).

  4. For the sake of completeness, one should assume that \(\gamma _0\) is not closable with respect to the norm topology of \({\mathsf {H}}\)—in order to ensure that \(D({\widehat{A}}_\Theta )\) is dense in \(\widehat{{\mathsf {H}}}\): This is the case in all applications.

References

  1. Aké, L., Flores, J.L., Sánchez, M.: Structure of globally hyperbolic spacetimes with timelike boundary. arXiv:1808.04412 [gr-qc]

  2. Amman, B., Große, N., Nistor, V.: Poincaré inequality and well-posedness of the Poisson problem on manifolds with boundary and bounded geometry. arXiv:1611.00281 [math-AP]

  3. Bachelot, A.: New boundary conditions on the time-like conformal infinity of the Anti-de Sitter universe. Comptes Rendus Math. 350, 359 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333, 1585 (2015). arXiv:1310.0738 [math-ph]

  5. Bär, C., Ginoux, N., Pfäffle, F.: Wave Equation on Lorentzian Manifolds and Quantization, p. 194. European Mathematical Society, Berlin (2007)

    Book  MATH  Google Scholar 

  6. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, 2nd edn, p. 635. CRC Press, New York (1996)

    MATH  Google Scholar 

  7. Behrndt, J.: Elliptic boundary value problems with \(\lambda \)-dependent boundary conditions. J. Differ. Equ. 249, 2663 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Behrndt, J., Langer, M.: Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples. In: de Snoo, H.S.V. (ed.) Operator Methods for Boundary Value Problems. London Mathematical Society Lecture Notes, p. 298. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  9. Benini, M., Dappiaggi, C.: Models of free quantum field theories on curved backgrounds. Chapter 3. In: Brunetti et al. (ed.) Advances in Algebraic Quantum Field Theory, Mathematical Physics Studies. Springer, Berlin (2015). arXiv:1505.04298 [math-ph]

  10. Benini, M., Dappiaggi, C., Hack, T.P.: Quantum field theory on curved backgrounds—a primer. Int. J. Mod. Phys. A 28, 1330023 (2013). arXiv:1306.0527 [gr-qc]

  11. Benini, M., Dappiaggi, C., Schenkel, A.: Algebraic quantum field theory on spacetimes with timelike boundary. Ann. Henri Poincare 19(8), 2401 (2018). arXiv:1712.06686 [math-ph]

  12. Chruściel, P.T., Galloway, G.J., Solis, D.: Topological censorship for Kaluza–Klein space–times. Ann. Henri Poincaré 10, 893 (2009). arXiv:0808.3233 [gr-qc]

  13. Dappiaggi, C., Ferreira, H.R.C.: On the algebraic quantization of a massive scalar field in anti-de-Sitter spacetime. Rev. Math. Phys. 30(02), 1850004 (2017). arXiv:1701.07215 [math-ph]

  14. Dappiaggi, C., Nosari, G., Pinamonti, N.: The Casimir effect from the point of view of algebraic quantum field theory. Math. Phys. Anal. Geom. 19(2), 12 (2016). arXiv:1412.1409 [math-ph]

  15. Dereziński, J., Siemssen, D.: An evolution equation approach to the Klein–Gordon operator on curved spacetime. arXiv:1709.03911 [math-ph]

  16. Derkach, V.A., Malamud, M.M.: Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eichhorn, J.: The Banach manifold structure of the space of metrics on noncompact manifolds. Differ. Geom. Appl. 1, 89 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feller, W.: Generalized second order differential operators and their lateral conditions. Ill. J. Math. 1, 459 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gal, C.G., Goldstein, G.R., Goldstein, J.A.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Große, N., Murro, S.: The well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary. arXiv:1806.06544 [math.DG]

  21. Große, N., Nistor, V.: Neumann and mixed problems on manifolds with boundary and bounded geometry. arXiv:1703.07228 [math-AP]

  22. Große, N., Schneider, C.: Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces. Math. Nachr. 286, 1586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grubb, G.: A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Sc. Norm. Super. Pisa 22(3), 425 (1968)

    MathSciNet  MATH  Google Scholar 

  24. Grubb, G.: Les problèmes aux limites généraux d’un opérateur elliptique provenant de la théorie variationnelle. Bull. Sci. Math. 94, 113–157 (1970)

    MathSciNet  MATH  Google Scholar 

  25. Grubb, G.: On coerciveness and semiboundedness of general boundary problems. Isr. J. Math. 10, 32 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hassi, S., Del Snoo, H., Szafraniec, F. (eds.): Operator Methods for Boundary Value Problems (London Mathematical Society Lecture Note Series). Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139135061

    Google Scholar 

  27. Hebey, E.: Sobolev Spaces on Riemannian manifolds, p. 113. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  28. Holzegel, G.: Well-posedness for the massive wave equation on asymptotically anti-de sitter spacetimes. J. Hyperb. Differ. Equ. 9, 239 (2012). arXiv:1103.0710 [gr-qc]

  29. Hörmander, L.: The Analysis of Linear Partial Differential Operators I, 2nd edn, p. 438. Springer, Berlin (1990)

    Google Scholar 

  30. Ibort, A., Lledó, F., Pérez-Pardo, J.M.: Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary. J. Funct. Anal. 268, 634 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ibort, A., Lledó, F., Pérez-Pardo, J.M.: On self-adjoint extensions and symmetries in quantum mechanics. Ann. Henri Poincare 16(10), 2367 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Lee, J.M.: Introduction to Smooth Manifolds, 2nd edn, p. 706. Springer, Berlin (2013)

    Google Scholar 

  33. Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications I. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  34. Malamud, M.: On a formula for the generalized resolvents of a non-densely defined Hermitian operator. Ukr. Math. J. 44, 1522 (1992)

    Article  MathSciNet  Google Scholar 

  35. Moretti, V.: Spectral Theory and Quantum Mechanics, 2nd edn, p. 950. Springer, Berlin (2018)

    MATH  Google Scholar 

  36. Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices 2, 483 (2008). arXiv:0703078 [math-ph]

  37. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, p. 400. Academic Press, London (1981)

    Google Scholar 

  38. Sanchez, M.: On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes. Differ. Geom. Appl. 24, 21 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Schick, T.: Manifolds with boundary and of bounded geometry. Math. Nachr. 233, 103 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Solis, D.A.: Global properties of asymptotically de Sitter and anti de Sitter spacetimes. Ph.D. thesis, University of Miami (2006). arXiv:1803.01171 [gr-qc]

  41. Triebel, H.: Theory of Function Spaces II, p. 370. Birkhauser, Basel (1992)

    Book  MATH  Google Scholar 

  42. Ueno, T.: Wave equation with Wentzell’s boundary condition and a related semigroup on the boundary, I. Proc. Jpn. Acad. 49, 672 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vasy, A.: The wave equation on asymptotically anti de sitter spaces. Anal. PDE 5, 81 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wrochna, M.: The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes. Lett. Math. Phys. 107, 2291 (2017). arXiv:1612.01203 [math-ph]

  45. Zahn, J.: Generalized Wentzell boundary conditions and quantum field theory. Ann. Henri Poinc. 19, 163 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of C. D. was supported by the University of Pavia. The work of N. D. was supported in part by a research fellowship of the University of Pavia. The work of N. D. and of H. F. was supported in part by a fellowship of the “Progetto Giovani GNFM 2017” under the project “Wave propagation on Lorentzian manifolds with boundaries and applications to algebraic QFT” fostered by the National Group of Mathematical Physics (GNFM-INdAM). We are grateful to Felix Finster, Nadine Grosse, Valter Moretti, Simone Murro and Juan Manuel Pérez-Pardo for the useful comments and discussions. We are grateful to Igor Khavkine for the useful comments, especially concerning Proposition 36.

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Authors and Affiliations

  1. Istituto Nazionale di Alta Matematica, Sezione di Pavia – Via Ferrata, 5, 27100, Pavia, Italy

    Claudio Dappiaggi, Nicoló Drago & Hugo Ferreira

  2. Dipartimento di Fisica, Università di Pavia, Via Bassi 6, 27100, Pavia, Italy

    Claudio Dappiaggi & Nicoló Drago

  3. INFN, Sezione di Pavia – Via Bassi 6, 27100, Pavia, Italy

    Hugo Ferreira

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  1. Claudio Dappiaggi
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Correspondence to Claudio Dappiaggi.

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Dappiaggi, C., Drago, N. & Ferreira, H. Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary. Lett Math Phys 109, 2157–2186 (2019). https://doi.org/10.1007/s11005-019-01173-z

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