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Kalyan B Sinha - One of the best experts on this subject based on the ideXlab platform.

Valentin A. Zagrebnov - One of the best experts on this subject based on the ideXlab platform.

  • A TrotterKato Product Formula for a Class of Non-Autonomous Evolution Equations, Trends in Nonlinear Analysis
    2008
    Co-Authors: Pierre-a. Vuillermot, Walter F. Wreszinski, Valentin A. Zagrebnov
    Abstract:

    In this article dedicated to Professor V. Lakshmikantham on the oc-casion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t; s)0stT generated by the sum (A(t) + B(t)) of two linear, time-dependent and generally unbounded operators de\u85ned on time-depen-dent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t; s)0stT as the strong limit in L(B) of a Product of the holomorphic contraction semigroups generated by A(t) and B(t), respectively, thereby getting a Product Formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t) +B(t)) to evolve with time provided there exists a \u85xed set D \t2[0;T] D(A(t) + B(t)) everywhere dense in B. We then mention several possible applica-tions of our Product Formula to various classes of non-autonomous par

  • trotter kato Product Formula and operator norm convergence
    Communications in Mathematical Physics, 1999
    Co-Authors: Hagen Neidhardt, Valentin A. Zagrebnov
    Abstract:

    We find necessary and sufficient conditions for the operator-norm convergence of the Trotter–Kato Product Formula. Using them we prove that this convergence takes place: (i) if the resolvent of one of the involved operators is compact, either (ii) if one operator is relatively compact with respect to another one, or (iii) if the Product of resolvents of the involved operators is compact.

  • on error estimates for the trotter kato Product Formula
    Letters in Mathematical Physics, 1998
    Co-Authors: Hagen Neidhardt, Valentin A. Zagrebnov
    Abstract:

    We study the error bound in the operator-norm topology for the Trotter exponential Product Formula as well as for its generalization a la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.

Takashi Ichinose - One of the best experts on this subject based on the ideXlab platform.

  • note on a Product Formula related to quantum zeno dynamics
    Annales Henri Poincaré, 2021
    Co-Authors: Pavel Exner, Takashi Ichinose
    Abstract:

    Given a nonnegative self-adjoint operator H acting on a separable Hilbert space and an orthogonal projection P such that $$H_P := (H^{1/2}P)^*(H^{1/2}P)$$ is densely defined, we prove that $$\lim _{n\rightarrow \infty } (P\,\mathrm {e}^{-itH/n}P)^n = \mathrm {e}^{-itH_P}P$$ holds in the strong operator topology. We also derive modifications of this Product Formula and its extension to the situation when P is replaced by a strongly continuous projection-valued function satisfying $$P(0)=P$$ .

  • results on convergence in norm of exponential Product Formulas and pointwise of the corresponding integral kernels
    arXiv: Mathematical Physics, 2009
    Co-Authors: Takashi Ichinose, Hideo Tamura
    Abstract:

    For the last one and a half decades it has been known that the exponential Product Formula holds also in norm in nontrivial cases. In this note, we review the results on its convergence in norm as well as pointwise of the integral kernels in the case for Schrodinger operators, with error bounds. Optimality of the error bounds is elaborated.

  • a Product Formula related to quantum zeno dynamics
    Annales Henri Poincaré, 2005
    Co-Authors: Pavel Exner, Takashi Ichinose
    Abstract:

    We prove a Product Formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space $\mathcal{H},$ with the convergence in $L_{{\text{loc}}}^2 (\mathbb{R};\mathcal{H}).$ It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P. The convergence in $\mathcal{H}$ is demonstrated in the case of a finite-dimensional P. The main result is illustrated in the example where the projection corresponds to a domain in $\mathbb{R}^d $ and the unitary group is the free Schrodinger evolution.

  • Product Formula related to quantum zeno dynamics
    arXiv: Mathematical Physics, 2003
    Co-Authors: Pavel Exner, Takashi Ichinose
    Abstract:

    We prove a Product Formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection $P$ on a separable Hilbert space $\HH$, with the convergence in $L^2_\mathrm{loc}(\mathbb{R};\HH)$. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace \hbox{$\mathrm{Ran} P$}. The convergence in $\HH$ is demonstrated in the case of a finite-dimensional $P$. The main result is illustrated in the example where the projection corresponds to a domain in $\mathbb{R}^d$ and the unitary group is the free Schr\"odinger evolution.

  • the norm convergence of the trotter kato Product Formula with error bound
    Communications in Mathematical Physics, 2001
    Co-Authors: Takashi Ichinose, Hideo Tamura
    Abstract:

    The norm convergence of the Trotter–Kato Product Formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative selfadjoint operators A and B which is selfadjoint.

Youness Lamzouri - One of the best experts on this subject based on the ideXlab platform.

  • a bias in mertens Product Formula
    International Journal of Number Theory, 2016
    Co-Authors: Youness Lamzouri
    Abstract:

    Rosser and Schoenfeld remarked that the Product ∏p≤x(1 − 1/p)−1 exceeds eγlogx for all 2 ≤ x ≤ 108, and raised the question whether the difference changes sign infinitely often. This was confirmed in a recent paper of Diamond and Pintz. In this paper, we show (under certain hypotheses) that there is a strong bias in the race between the Product ∏p≤x(1 − 1/p)−1 and eγlogx which explains the computations of Rosser and Schoenfeld.

  • A bias in Mertens' Product Formula
    arXiv.org, 2014
    Co-Authors: Youness Lamzouri
    Abstract:

    Rosser and Schoenfeld remarked that the Product \prod_{p\leq x}(1-1/p)^{-1} exceeds e^{γ} \log x for all 2\leq x\leq 10^8, and raised the question whether the difference changes sign infinitely often. This was confirmed in a recent paper of Diamond and Pintz. In this paper, we show (under certain hypotheses) that there is a strong bias in the race between the Product \prod_{p\leq x}(1-1/p)^{-1} and e^{γ}\log x which explains the computations of Rosser and Schoenfeld.

Biswarup Das - One of the best experts on this subject based on the ideXlab platform.