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

Abstract

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.

References in corpus (1)

• Spectral and scattering theory for space-cutoff $P(\phi {)}_2$ models with variable metric

Cited by in corpus (5)

• On eigenfunction expansion of solutions to the Hamilton equations
• Dynamical instabilities and quasi-normal modes, a spectral analysis with applications to black-hole physics
• Quantum field theory in static external potentials and Hadamard states
• Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation
• High-Momenta Estimates for the Klein-Gordon Equation: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering