P. Exner, H. Neidhardt
Trotter-Kato product formula for unitary groups
(296K, pdf)
ABSTRACT. Let $A$ and $B$ be non-negative self-adjoint operators in a separable
Hilbert space such that its form sum $C$ is densely defined. It is shown
that the Trotter product formula holds for imaginary times in the
$L^2$-norm, that is, one has
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\begin{displaymath}
\lim_{n\to+\infty}\int^T_0 \left\|\left(e^{-itA/n}e^{-itB/n}\right)^nh -
e^{-itC}h\right\|^2dt = 0
\end{displaymath}
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for any element $h$ of the Hilbert space and any $T > 0$. The result
remains true for the Trotter-Kato product formula
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\begin{displaymath}
\lim_{n\to+\infty}\int^T_0 \left\|\left(f(itA/n)g(itB/n)\right)^nh -
e^{-itC}h\right\|^2dt = 0
\end{displaymath}
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where $f(\cdot)$ and $g(\cdot)$ are so-called holomorphic
Kato functions; we also derive a canonical representation for any function of this class.