Wigner measures supported on weak KAM tori

Parmeggiani, Alberto; Zanelli, Lorenzo

doi:10.1007/s11854-014-0015-8

Cited by 13 publications

(16 citation statements)

References 34 publications

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“…In fact, as shown by Proposition 2.3 in [27], the Weyl quantizations as in (2.3) and (2.10) coincide.…”

Section: Remark 22 Inmentioning

confidence: 69%

“…This means that it is a well-defined positive function on the torus. The function (4.21) fulfills (see Proposition 4.6 in [27])…”

Section: Remark 44 (Example)mentioning

confidence: 96%

“…In [27] a class of WKB states on the torus with regularized phase function have been defined in such a way that the associated Wigner measures are coinciding with the Legendre transform of the so-called Mather measures.…”

Section: Introductionmentioning

confidence: 99%

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On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

Paul

Zanelli

2014

J Fourier Anal Appl

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In the framework of toroidal Pseudodifferential operators on the flat torus T n := (R/2π Z) n we begin by proving the closure under composition for the class ofwe exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on T n × R n written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space H 1 (T n ; C) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton-Jacobi equation 12 |P + ∇ x v(P, x)| 2 + V (x) =H (P) for P ∈ Z n with > 0, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of P + ∇ x v.

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“…In fact, as shown by Proposition 2.3 in [27], the Weyl quantizations as in (2.3) and (2.10) coincide.…”

Section: Remark 22 Inmentioning

confidence: 69%

“…This means that it is a well-defined positive function on the torus. The function (4.21) fulfills (see Proposition 4.6 in [27])…”

Section: Remark 44 (Example)mentioning

confidence: 96%

See 1 more Smart Citation

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

Paul

Zanelli

2014

J Fourier Anal Appl

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“…The toroidal quantization has since led to many further developments and applications, see e.g. [LNJP16,PZ14b,PZ14a,Car14], to mention a few. So, the described link leads to a way of transferring results from the toroidal setting to the lattice.…”

Section: Relation Between Lattice and Toroidal Quantizationsmentioning

confidence: 99%

Difference equations and pseudo-differential operators onZn

Botchway

Kibiti

Ruzhansky

2020

Journal of Functional Analysis

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In this paper we develop the calculus of pseudo-differential operators on the lattice Z n , which we can call pseudo-difference operators. An interesting feature of this calculus is that the phase space is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable. We establish formulae for composition, adjoint, transpose, and for parametrix for the elliptic operators. We also give conditions for the ℓ 2 , weighted ℓ 2 , and ℓ p boundedness of operators and for their compactness on ℓ p . We describe a link to the toroidal quantization on the torus T n , and apply it to give conditions for the membership in Schatten classes on ℓ 2 (Z n ). Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their ℓ 2 -boundedness. The results are applied to give estimates for solutions to difference equations on the lattice Z n .Contents 1991 Mathematics Subject Classification. 58J40, 35S05, 35S30, 42B05, 47G30.

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“…We underline that a complete study of the link between the effective Hamiltonian, viscosity solutions of Hamilton-Jacobi equation and Schrödinger eigenvalue problem should also involve the phase space analysis of eigenfunctions (and energy quasimodes). In this direction, some preliminary results for the n-dim case have been obtained [18][19][20]. For the time evolution of WKB-type wave functions, we address the reader to the works [21][22][23] (and references therein).…”

Section: Introductionmentioning

confidence: 99%