On the Inverse Problem of a Dissipative Scattering Theory. I

Abstract

Let L be a selfadjoint operator on the separable Hilbert space L. By H we denote a maximal dissipative operator,

$$\operatorname{Im} \left( {Hf,f} \right) \leqslant 0,$$

(1.1)

f ε dom H, on a separable Hilbert space G. The operator H generates a one-parameter contraction semigroup T(t),

$$T\left( t \right) = {e^{ - ith}},$$

(1.2)

t ≥ 0. By J we denote a bounded operator from L into G. The wave operator W₊(H*,L;J) is defined by

$${W_ + }\left( {{H^*},L;J} \right) = \mathop {s - \lim }\limits_{t \to + \infty } \,{e^{it{H^*}}}J{e^{ - itL}}{P_{ac}}\left( L \right),$$

(1.3)

where P_ac(L) denotes the projection onto the absolutely continuous subspace Lac of the self adjoint operator L. Similarly we introduce the wave operator W_-(H,L;J),

$${W_ - }\left( {H,L;J} \right) = \mathop {s - \lim }\limits_{t \to + \infty } \,{e^{ - itH}}J{e^{itL}}{P_{ac}}\left( L \right).$$

(1.4)