On the absence of necessary conditions for linear evolution operators

Abstract

There exist selfadjoint operators A ( t ) ( t ⩾ 0 ) A(t)(t \geqslant 0) , whose resolvents depend smoothly on t, such that the initial value (Schrödinger) problem d u ( t ) / d t = i A ( t ) u ( t ) , u ( 0 ) = f du(t)/dt = iA(t)u(t),u(0) = f has a unique ( C ∞ ) ({C^\infty }) solution u for each f in a dense set in the underlying Hilbert space, with u depending continuously on f, and yet the intersection over t ⩾ 0 t \geqslant 0 of the domain of A ( t ) A(t) is {0}.