06-268 Asao Arai
Heisenberg Operators, Invariant Domains and Heisenberg Equations of Motion (65K, Latex2.09) 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 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.

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