Аннотация:
Since 1875 due to Sophus Lie it is known that for any pair of
(noncommutative) finite
square matrices $A$ and $B$ as generators one has the norm estimate $O(1/n)$
for convergence rate
of the exponential product formula. In 1959 H.Trotter proved this
formula in the strong
operator topology on the Banach space for strongly continuous
semigroups and unbounded
generators $A$ and B. Further, in 1978 T. Kato extended this result
(still in the strong operator
topology) to the non-exponential product formulae.
A breakthrough result in this direction was presented in the Dzh. L. Rogava theorem (1993).
It says that on a separable Hilbert space the exponential Trotter
product formula may
converge in the operator-norm topology with convergence rate of the
order $O(\ln{n}/\sqrt{n})$.
This discovery initiated a number of papers addressed to the study of
conditions on generators A
and B aiming to optimise the rate of convergence in Rogava’s assertion.
Motivated by this discovery the optimal rate of convergence $O(1/n)$ in
the operator-norm
topology under conditions of the Rogava theorem was proved only in
2001 (the Ichinose-Tamura-Tamura-Zagrebnov theorem) for both the Trotter and the
Trotter-Kato product formulae.
Under new fractional conditions on generators A and B the optimal rate
of the Trotter-Kato product
formulae convergence in the operator-norm topology on a Hilbert space
was established in the Ichinose-Neidhardt-Zagrebnov (INZ)-theorem (2004).
I shall present these and some other recent results about the
Lie-Trotter-Kato product formulae
on Hilbert and Banach spaces, which are collected in the book:
V. A. Zagrebnov, H. Neidhardt, T. Ichinose, Trotter-Kato Product Formulae, 2024.
Язык доклада: английский
|
