Scattering theory for Klein-Gordon equations with non-positive energy
Résumé
We study the scattering theory for charged Klein-Gordon equations: \[ \left\{ \begin{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x)-\epsilon^{2}(x, D_{x})\phi(t,x)=0,\\[2mm] \phi(0, x)= f_{0}, \\[2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, \end{array}\right. \] where: \[ \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}\left(\p_{x_{j}}-\i b_{j}(x)\right)A^{jk}(x)\left(\p_{x_{k}}-\i b_{k}(x)\right)+ m^{2}(x), \] 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: \[ h[f, f]:= \int_{\rr^{n}}\overline{f}_{1}(x) f_{1}(x)+ \overline{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \overline{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. \] 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-dep\-endent 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.
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