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

  • Claudio DappiaggiEmail author
  • Nicoló Drago
  • Hugo Ferreira


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 (Mg). The latter is diffeomorphic to any constant time hypersurface of the underlying background. In turn, assuming that (Mg) 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.


Fundamental solutions Globally hyperbolic spactimes with timelike boundary Boundary triples 

Mathematics Subject Classification

65N80 81T20 



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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Istituto Nazionale di Alta MatematicaPaviaItaly
  2. 2.Dipartimento di FisicaUniversità di PaviaPaviaItaly
  3. 3.INFNPaviaItaly

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