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Scattering theory for Klein-Gordon equations with non-positive energy

We study the scattering theory for charged Klein-Gordon equations: \[\{{array}{l} (\p_{t}- ıv(x))^{2}ϕ(t,x) ε^{2}(x, D_{x})ϕ(t,x)=0,[2mm] ϕ(0, x)= f_{0}, [2mm] ı^{-1} \p_{t}ϕ(0, x)= f_{1}, {array}. \] where: \[ε^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} ıb_{j}(x))A^{jk}(x)(\p_{x_{k}} ıb_{k}(x))+ m^{2}(x),\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \[ h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)ε^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. \] We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case.