Time delay is a common feature of quantum scattering theory

Richard, Serge; Aldecoa, Rafael Tiedra de

doi:10.1016/j.jmaa.2011.09.021

Cited by 11 publications

(37 citation statements)

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“…This property corresponds in the quantum case to the fact that the time delay operator is decomposable in the spectral representation of the free Hamiltonian (see [29,Remark 4.4]). In the next corollary, we show that the unsymmetrized time delay…”

Section: Time Delay In Classical Scattering Theorymentioning

confidence: 99%

“…exists and is equal to the symmetrized time delay in the limit r → ∞ if the scattering process preserves the norm of the velocity vector ∇H 0 (the superscript "in", borrowed from [29,Sec. 4], refers to "incoming" time delay).…”

Section: Time Delay In Classical Scattering Theorymentioning

confidence: 99%

“…Our interest in these issues has been aroused by recent articles on time delay in quantum mechanics [28,29] and on the scattering matrix in Hamiltonian mechanics [8]. In [29], the authors prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay [33,35], is a common feature of two-Hilbert spaces quantum scattering theory.…”

Section: Introductionmentioning

confidence: 99%

“…In [29], the authors prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay [33,35], is a common feature of two-Hilbert spaces quantum scattering theory. Their proofs rely on abstract commutator methods and on an integral formula relating localization operators to time operators [28].…”

Section: Introductionmentioning

confidence: 99%

“…4] that there exist natural position observables Φ satisfying (1.1) for many Hamiltonian systems (M, H 0 ) appearing in literature. Second, we know that this approach works in the quantum case [29]. Finally, the condition (1.1) is formulated in an invariant way on M , without any mention to the particular structure of H 0 .…”

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Time Delay and Calabi Invariant in Classical Scattering Theory

Gournay

Aldecoa

2012

Rev. Math. Phys.

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We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H 0 , H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of Buslaev and Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincaré ball.In consequence, the time evolution of the observables Φ j under the flow ϕ 0is well chosen, the perturbed trajectory corresponding to the free trajectory {ϕ 0 t (q, p)} t∈R also escapes from each ball B r as |t| → ∞. In such a case, the difference of sojourn times in B r between the two trajectories may converge to a finite value, called the global time delay for (q, p), as r → ∞. This is well known and has been established by various authors for different types of perturbations V (see, for instance, [5-7, 12, 20, 26, 27, 34]). Fine. But what happens when H 0 and H are abstract Hamiltonians on a given symplectic manifold M ? If the dimension of M is finite, Darboux's theorem guarantees us that there exist, at least locally, canonical coordinates on M . However, these coordinates have usually nothing to do with the free Hamiltonian H 0 , and are in consequence inappropriate for the definition of sojourn times. Therefore, our point of view is instead to retain as position observables merely functions Φ j satisfying (1.1), as in the case H 0 (q, p) = |p| 2 /2. This choice is certainly not the most general one, but it turns out to be extremely rewarding as we shall explain below. Here, we just note three facts to its favor. First, it has been shown in [14, Sec. 4] that there exist natural position observables Φ satisfying (1.1) for many Hamiltonian systems (M, H 0 ) appearing in literature. Second, we know that this approach works in the quantum case [29]. Finally, the condition (1.1) is formulated in an invariant way on M , without any mention to the particular structure of H 0 . So, let H 0 and H be Hamiltonians on a symplectic manifold M with Poisson bracket {·, ·}, assume that H 0 and H have complete flows {ϕ 0 t } t∈R and {ϕ t } t∈R , and let Φ := (Φ 1 , . . . , Φ d ) be a family of observables satisfying (1.1). Then, the vector ∇H 0 := ({Φ 1 , H 0 }, . . . , {Φ d , H 0 }) and the set Crit(H 0 , Φ) := (∇H 0 ) −1 ({0}) ⊂ M can be interpreted, respectively, as the velocity observable and the set of critical points associated to H 0 and Φ (see [28, Assumption 2.2 and Definition 2.5] 1250023-2 Rev. Math. Phys. 2012.24. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/05/15. For personal use only.

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Section: Time Delay In Classical Scattering Theorymentioning

confidence: 99%

Section: Time Delay In Classical Scattering Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Time Delay and Calabi Invariant in Classical Scattering Theory

Gournay

Aldecoa

2012

Rev. Math. Phys.

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Time Delay for the Dirac Equation

Naumkin

Weder

2016

Lett Math Phys

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We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operatorHere H 0 is the free Dirac operator and ζ (t) is such that ζ (t) = 1 for 0 ≤ t ≤ 1 and ζ (t) = 0 for t > 1. This approach allows us to obtain the time delay operator δT (f ) for initial states f in H 3/2+ε 2 R 3 ; C 4 , ε > 0, the Sobolev space of order 3/2 + ε and weight 2. The relation between the time delay operator δT (f ) and the Eisenbud-Wigner time delay operator is given. Also, the relation between the averaged time delay and the spectral shift function is presented.

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A theoretical investigation of elastic scattering of H atom by C₆₀ and Kr@C₆₀

Dubey¹,

Agrawal²,

Rao³

et al. 2019

J. Phys. B: At. Mol. Opt. Phys.

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We report an extensive study of cross section, phase shift and time delay for H–Kr@C60 elastic scattering. To demonstrate the effect of encapsulation, we extend this study to H–C60 elastic scattering also. Interaction potential is calculated using density functional theory for the scattering through both hexagonal and pentagonal rings of bare C60 and Kr@C60. The interaction potential for H–C60 is a double-humped barrier. In H–Kr@C60 case, this interaction has an additional huge core repulsive region near the center of C60, which is due to presence of the encapsulated atom. The scattering cross section exhibits glory oscillations and resonances, which are qualitatively different for the scattering through pentagonal and hexagonal rings of C60. The observed resonances arise due to combined effect of centrifugal and C60 potential barrier. We notice bunching of phase shifts for even–even and odd–odd ℓ partial waves in H–C60 scattering, consequently this happens in all measured scattering observables. The presence of the confined atom eliminates this phase bunching; therefore, H–Kr@C60 scattering rather gives well-organized and sequential placement of resonances in order of ℓ values.

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Time delay is a common feature of quantum scattering theory

Abstract: International audienceWe prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay, is a common feature of two-Hilbert spaces quantum scattering theory. All statements are model-independent

Cited by 11 publications

References 26 publications

Time Delay and Calabi Invariant in Classical Scattering Theory

Time Delay and Calabi Invariant in Classical Scattering Theory

Time Delay for the Dirac Equation

A theoretical investigation of elastic scattering of H atom by C₆₀ and Kr@C₆₀

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Time delay is a common feature of quantum scattering theory

Abstract: International audienceWe prove that the existence of time delay defined in terms of sojourn times, as well as its identity with Eisenbud-Wigner time delay, is a common feature of two-Hilbert spaces quantum scattering theory. All statements are model-independent

Cited by 11 publications

References 26 publications

Time Delay and Calabi Invariant in Classical Scattering Theory

Time Delay and Calabi Invariant in Classical Scattering Theory

Time Delay for the Dirac Equation

A theoretical investigation of elastic scattering of H atom by C60 and Kr@C60

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Product

Resources

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A theoretical investigation of elastic scattering of H atom by C₆₀ and Kr@C₆₀