Asao Arai
 Heisenberg Operators, Invariant Domains and Heisenberg Equations of Motion
(65K, Latex2.09)

ABSTRACT.  An abstract operator theory is developed 
on operators of the form 
$A_H(t):=e^{itH}Ae^{-itH}, \, t\in \R$, with 
$H$ a self-adjoint operator and $A$ a 
linear operator on a Hilbert space (in the context of quantum mechanics, 
$A_H(t)$ is called the Heisenberg 
operator of $A$ with respect to $H$). 
The following aspects are discussed: 
(i) integral equations for $A_H(t)$ for a general class of $A$ ; 
(ii) a sufficient condition for $D(A)$, the domain of $A$, to be left invariant 
by $e^{-itH}$ for all $t \in \R$ ; 
(iii) a mathematically rigorous formulation of the Heisenberg equation of motion 
in quantum mechanics and 
 the uniqueness of its solutions ; (iv) invariant domains in the case where $H$ is 
an abstract version 
of Schr\"odinger and Dirac operators ; (v) applications to Schr\"odinger 
operators with matrix-valued potentials and Dirac operators.