Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques

Robert, Didier; Helffer, Bernard

doi:10.5802/aif.844

Cited by 38 publications

(4 citation statements)

References 11 publications

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“…Helffer studied this operator to discuss the operator −h 2 Δ + VðxÞ associated with the parameter h, where Δ is the Laplace operator and VðxÞ denotes some potential function for any x ∈ ℝ n . Moveover, the h-pseudodifferential operator also provides a rigorous way to establish relationship each other for quantum physics and classical mechanics; see, for example, [2,3]. Furthermore, by using h-pseudodifferential operator, the Cauchy problem of semiclassical elliptic partial differential equations is studied in [3][4][5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators

Yang

2021

Journal of Function Spaces

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Let T a h be the h -pseudodifferential operators with symbol a . When a ∈ S ρ , 1 m and m = n ρ − 1 / 2 , it is well known that T a h is not always bounded in L 2 ℝ n . In this paper, under the condition a x , ξ ∈ L ∞ S ρ n ρ − 1 / 2 ω , we show that T a h is bounded on L 2 .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators

Yang

2021

Journal of Function Spaces

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“…Using the Tauberian method, the Weyl law for semiclassical Schrödinger operator (1.3) was obtained in [70], with a remainder O( ), under the assumption that ∇V = 0 along the whole edge. Asymptotics on the diagonal in the bulk and at a non-degenerate edge (that is, Theorems II.1 and II.2 at x = y = 0) have been computed in [56] using the preliminary results of [20], under a growth condition of V at infinity.…”

Section: Semiclassical Projector Asymptoticsmentioning

confidence: 99%

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Deleporte¹,

Lambert²

2021

Preprint

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We study local asymptotics for the spectral projector associated to a Schrödinger operator − 2 ∆ + V on R n in the semiclassical limit as → 0. We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply for a general class of smooth potentials in arbitrary dimension n ≥ 1. We also obtain a central limit theorem which characterizes the mesoscopic fluctuations of these Fermi gases in the bulk.

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“…The operators I h th/fr (t) belong to the class of Fourier integral operators (FIO). Designing operators that approximate the dynamics of a semi-classical propagator goes back to the early days of semi-classical analysis, see J. Chazarain [19], B. Helffer and D. Robert [47] and [81], see also the books [93,Chapter 12] or [25].…”

Section: Semi-classical Approximation Of the Propagatormentioning

confidence: 99%

Semi-classical analysis on H-type groups

Kammerer

Fischer

2019

Sci. China Math.

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In this paper, we develop a semi-classical analysis on H-type groups. We define semiclassical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consists of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensioned representations of the group, even though they are negligible with respect to the Plancherel measure.

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Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques

Cited by 38 publications

References 11 publications

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Semi-classical analysis on H-type groups

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Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques

Cited by 38 publications

References 11 publications

On L 2 -Boundedness of h -Pseudodifferential Operators

On L 2 -Boundedness of h -Pseudodifferential Operators

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

Semi-classical analysis on H-type groups

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Product

Resources

About

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators

On $L^{2}$ -Boundedness of $h$ -Pseudodifferential Operators