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An evolution equation approach to the Klein-Gordon operator on curved spacetime. (English) Zbl 1423.58016

The authors develop a theory for the Klein-Gordon equation on curved spacetimes. Precisely, they consider the Klein-Gordon operator on a Lorentzian manifold coupled to an electromagnetic potential \(A\) and a scalar potential \(Y\): \[K = |g|^{-1/2} (D_{\mu} - A_{\mu} )|g|^{1/2} g^{\mu \nu} (D_{\nu} -A_{\nu}) +Y\] where \(|g|= |\det[ g_{\mu \nu}]|\) and \(D_{\mu}= -i \partial_{\mu}\). Both classical propagators and some families of nonclassical propagators are constructed. The theory of nonautonomous evolution equations on Hilbert spaces is used to this end. Assumptions on the metric and potentials are weak, but global in spacetime. In an Appendix, the authors illustrate how some of the assumptions can be realized.

MSC:

58J45 Hyperbolic equations on manifolds
35L05 Wave equation
47D06 One-parameter semigroups and linear evolution equations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81T20 Quantum field theory on curved space or space-time backgrounds
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