 06268 Asao Arai
 Heisenberg Operators, Invariant Domains and Heisenberg Equations of Motion
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Sep 25, 06

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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 selfadjoint 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 matrixvalued potentials and Dirac operators.
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