Time boundedness of the energy for the charge transfer model (Q917803)

The Schrödinger equation \(id\psi /dt=H(t)\psi\) in the space \(H^ 2({\mathbb{R}}^{\nu})\) is investigated where \(H(t)=-\Delta /2m+V(t)\), \(V(t)=\sum^{N}_{1}V_ j(x-y_ j(t))\), with the classical trajectories \(y_ j(t)\) given in advance and satisfying \(d^ 2y_ j/dt^ 2\in L_ 1({\mathbb{R}})\), \(| y_ j(t)-y_ k(t)| \geq \alpha t\) \((t>const.\), \(\alpha >0\), \(j\neq k)\). The potential V is subjected to certain weak requirements (which cannot be stated here), in particular the gradients \(\nabla V_ j\) are assumed to be bounded and integrable outside a compact set. Result: the time evolution \(\psi\) (t) has a uniformly bounded energy expectation, \(\sup (\psi (t),H(t)\psi (t))<\infty\), \(t>const\).