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 
% 
% 
\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} 
% 
% 
for any element $h$ of the Hilbert space and any $T > 0$. The result 
remains true for the Trotter-Kato product formula 
% 
% 
\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} 
% 
% 
where $f(\cdot)$ and $g(\cdot)$ are so-called holomorphic 
Kato functions; we also derive a canonical representation for any function of this class.