Time operators and shift representation of dynamical systems

Antoniou, Ioannis; Садовничий, В. А.; Shkarin, Stanislav

doi:10.1016/s0378-4371(99)00070-9

Cited by 15 publications

(8 citation statements)

References 10 publications

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“…The range of J consists of absolutely continuous functions and is, of course, dense in C [0,1] . Since the series (31) is also uniformly convergent [12], we can apply J to both sides getting…”

Section: Time Operators On Function Spacesmentioning

confidence: 99%

“…These sufficient conditions involve the notion of moduli of continuity which we want to recall. For g ∈ C [0,1] its modulus of continuity ω 1 g and the modulus of continuity of the second order ω 2 g are defined as follows: 1] |x−y| δ g(x) − g( y) , 0 δ 1, ω (2) g (δ) = sup…”

Section: Time Operators On Function Spacesmentioning

confidence: 99%

“…Later on time operators have been also defined on the spaces L 2 associated with exact dynamical systems [2] and, in a more general setting, associated with unilateral shifts on abstract Hilbert spaces [1]. The question of existence of selfadjoint time operators for unitary evolutions has been recently revisited in [10,11] on the basis of Halmos-Helson theory of invariant subspaces, Sz.-Nagy-Foia¸s dilation theory and MisraPrigogine-Courbage (MPC) theory of irreversibility, including an extensive set of equivalent conditions written in terms of Schrödinger couples, the Weyl commutation relation, incoming and outgoing subspaces, innovation processes, Lax-Phillips scattering, translation and spectral representations, and spectral properties.…”

Section: Introductionmentioning

confidence: 99%

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Evolution semigroups and time operators on Banach spaces

Suchanecki

Gómez-Cubillo

2010

Journal of Mathematical Analysis and Applications

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We present new general methods to obtain shift representation of evolution semigroups defined on Banach spaces. We introduce the notion of time operator associated with a generalized shift on a Banach space and give some conditions under which time operators can be defined on an arbitrary Banach space. We also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on Banach spaces satisfying some geometric conditions. The last part of the paper contains some examples of explicit constructions of time operators on function spaces.

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Section: Time Operators On Function Spacesmentioning

confidence: 99%

Section: Time Operators On Function Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Evolution semigroups and time operators on Banach spaces

Suchanecki

Gómez-Cubillo

2010

Journal of Mathematical Analysis and Applications

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“…Fortunately, the Koopman operator provides such a tool that drives the evolution of functions (also called observables) defined in a phase space [9] and thus shifts the focus to the ensemble of trajectories. In the early days, theoretical aspects of this operator were discussed in statistical or mechanical systems [10][11][12]. In recent years, a surge of Koopman analysis and its applications pervade diverse fields, such as power systems [13][14][15][16], building energy efficiency models [17], fluid systems [18][19][20][21], chemistry [22], robotics [23][24][25] and biology [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Koopman analysis of nonlinear systems with a neural network representation

Lan

2022

Commun. Theor. Phys.

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The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields. The intrinsic complexity of their dynamics defies many existing tools based on individual orbits, while Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits, which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator. However, it is difficult to identify and represent the most relevant eigenfunctions in practice. Here, combined with the Koopman analysis, a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems. By invoking the error minimization, a fundamental set of Koopman eigenfunctions are derived, which may reproduce the input dynamics through a nonlinear transformation provided by the neural network. The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.

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“…Time operators have been introduced for Liouville quantum dynamics [37,10,11], relativistic fields [4], unilateral shifts [6], diffusions [5], semigroups with normal generators [45] or dilations of Markov processes [7]. Also the connections of the time operators with innovation processes [54] and Lax-Phillips scattering processes [31] have been repeatedly emphasized as in [21,3,2,49] and [35,4,6,2], respectively.…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint time operators and invariant subspaces

Gómez¹

2008

Reports on Mathematical Physics

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For classical dynamical systems time operators are introduced as selfadjoint operators satisfying the so called weak Weyl relation (WWR) with the unitary groups of time evolution. Dynamical systems with time operators are intrinsically irreversible because they admit Lyapounov operators as functions of the time operator. For quantum systems selfadjoint time operators are defined in the same way or as dilations of symmetric ones dealing with the time-energy uncertainty relation, times of occurrence or survival probabilities. This work tackles the question of the existence of selfadjoint time operators on the basis of the Halmos-Helson theory of invariant subspaces, the Sz.-Nagy-Foiaş dilation theory and the Misra-Prigogine-Courbage theory of irreversibility. It is shown that the existence of a selfadjoint time operator for a unitary evolution is equivalent to the intrinsic irreversibility of the evolution plus the existence of a simply invariant subspace or a rigid operator-valued function for its Sz.-Nagy-Foiaş functional model. An extensive set of equivalent conditions to the existence of selfadjoint time operators can be obtained from these results. Such conditions are written in terms of Schrödinger couples, the Weyl commutation relation, incoming and outgoing subspaces, innovation processes, Lax-Phillips scattering processes, and translation and spectral representations. As an example, the selfadjoint extension of the quantum Aharonov-Bohm time-of-arrival operator is studied.

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Time operators and shift representation of dynamical systems

Cited by 15 publications

References 10 publications

Evolution semigroups and time operators on Banach spaces

Evolution semigroups and time operators on Banach spaces

Koopman analysis of nonlinear systems with a neural network representation

Self-adjoint time operators and invariant subspaces

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