Born-Jordan pseudodifferential operators with symbols in the Shubin classes

Cordero, Elena; Gosson, Maurice de; Nicola, Fabio

doi:10.1090/btran/16

Cited by 4 publications

(5 citation statements)

References 28 publications

(42 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have proven in a recent work [9] with E. Cordero that the solution b actually exists in S ′ (R 2n ), but the method is quite tricky and does not allow an explicit expression of b, neither does it allow to produce any qualitative results about the regularity properties of b in terms of those of a. However, as we have shown in [10], the situation is much more satisfactory when one supposes that the symbol a belongs to one of the Shubin symbol classes Γ m ρ (R 2n ). One has in this case the following result, which in a sense trivializes Born-Jordan operators:…”

Section: Born-jordan Operatorsmentioning

confidence: 96%

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Gosson

Nicola

2018

Bulletin des Sciences Mathématiques

Self Cite

View full text Add to dashboard Cite

Quantization procedures play an essential role in microlocal analysis, time-frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, a(x, ξ) an operator Op(a), according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol a(x, ξ) = f (x) depends only on x or a(x, ξ) = g(ξ) depends only on ξ:where F stands for the Fourier transform. Now, Dirac aimed at finding a quantization satisfying, in addition, the key correspondence [Op(a), Op(b)] = iOp({a, b})where [ , ] stands for the commutator and { , } for the Poisson brackets, which would represent a tight link between classical and quantum * maurice.de.gosson@univie.ac.at † fabio.nicola@polito.it mechanics. Unfortunately, the famous Groenewold-van Hove theorem states that such a quantization does not exist, and indeed most quantization rules satisfy this property only approximately. Now, in this note we show that the above commutator rule in fact holds for the Born-Jordan quantization, at least for symbols of the type f (x) + g(ξ). Moreover we will prove that, remarkably, this property completely characterizes this quantization rule, making it the quantization which best fits the Dirac dream.Op BJ (a) = 1 0 Op τ (a)dτ. Now, for a given quantization rule a −→ Op(a), regarded as a mapping S ′ (R 2n ) −→ L(S(R n ), S ′ (R n )),

show abstract

Section: Born-jordan Operatorsmentioning

confidence: 96%

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Gosson

Nicola

2018

Bulletin des Sciences Mathématiques

Self Cite

View full text Add to dashboard Cite

show abstract

“…that is to say, the symbol b 0 (t, z) is in the Shubin class 0 1 (R 2d ) with uniform estimates w.r. For applications of Shubin classes in the framework of Born-Jordan quantization we refer to the work [7]. (16) It can be used to define the τ -pseudodifferential operator with symbol σ via the formula…”

Section: Definition 23 Let (A J ) J Be a Sequence Of Symbols A Jmentioning

confidence: 99%

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

Cordero

2021

J. Pseudo-Differ. Oper. Appl.

Self Cite

View full text Add to dashboard Cite

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $$H^\beta =(-\Delta +|x|^2)^\beta $$ H β = ( - Δ + | x | 2 ) β , $$0<\beta \le 1$$ 0 < β ≤ 1 . We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class $$S^m_{0,0}$$ S 0 , 0 m , $$m\in \mathbb {R}$$ m ∈ R .

show abstract

“…As mentioned in the Introduction, for any given time-frequency representation Q σ = σ * Wig we can consider, by formula (3), the operator T a σ depending on the symbol a. The class of operators that we obtain contains classical pseudodifferential operators, localization, Weyl and τ -Weyl operators, see [6], as well as many other kind of operators of pseudodifferential type, such as the ones associated with the Born-Jordan representation, see [12], and the pseudo-differential operators considered in [1]. The aim of this section is to introduce some basic facts about the correspondence a → T a σ and establish a Lebesgue functional setting where these operators act continuously.…”

Section: Cohen Operators: Quantizations and Boundednessmentioning

confidence: 99%

Cohen class of time-frequency representations and operators: Boundedness and uncertainty principles

Boggiatto

Carypis

Oliaro

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

This paper presents a proof of an uncertainty principle of Donoho-Stark type involving ε-concentration of localization operators. More general operators associated with time-frequency representations in the Cohen class are then considered. For these operators, which include all usual quantizations, we prove a boundedness result in the L p functional setting and a form of uncertainty principle analogous to that for localization operators.

show abstract

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

Cited by 4 publications

References 28 publications

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

Cohen class of time-frequency representations and operators: Boundedness and uncertainty principles

Contact Info

Product

Resources

About