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

We study the scattering theory for charged Klein-Gordon equations: $$\{arrayl({\text{p}}_t-\text{i}v(x){)}^2\varphi (t,x){ϵ}^2(x,D_x)\varphi (t,x)=0,[2mm]\varphi (0,x)=f_0,[2mm]{\text{i}}^{-1}{\text{p}}_t\varphi (0,x)=f_1,array\mathrm{.}$$ where: $${ϵ}^2(x,D_x)=\sum _{1\le j,k\le n}({\text{p}}_{x_j}\text{i}{b}_j(x))A^{jk}(x)({\text{p}}_{x_k}\text{i}{b}_k(x))+m^2(x),$$ 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: $$h[f,f]:={\int }_{{\text{rr}}^n}{\bar{f}}_1(x)f_1(x)+{\bar{f}}_0(x){ϵ}^2(x,D_x)f_0(x)-{\bar{f}}_0(x)v^2(x)f_0(x)\text{d}x\mathrm{.}$$ 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.