Characterization of Smooth Symbol Classes by Gabor Matrix Decay

Bastianoni, Federico; Cordero, Elena

doi:10.1007/s00041-021-09895-2

Cited by 6 publications

(5 citation statements)

References 39 publications

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“…Since μ(χ) −1 = μ(χ −1 ) is in F I O(χ −1 , q, v s ) by Theorem 6.8, the algebra property of Theorem 6.4 (ii) implies that T μ(χ −1 ) ∈ F I O(Id, q, v s ). Now Theorem 3.2 in [2] implies the existence of a symbol σ 1 ∈ M ∞,q 1⊗v s (R 2d ), such that T μ(χ) −1 = Op w (σ 1 ), as claimed. The rest goes exactly as in [9,Theorem 3.8].…”

Section: Theorem 63 Consider T a Continuous Linear Operatormentioning

confidence: 73%

“…It follows the same pattern of the proof of [9,Theorem 3.8]. The main tool is the characterization in Theorem 3.2 of [2] which extends Theorem 4.6 in [25] to the case 0 < q < 1. We recall the main steps for the benefit of the reader.…”

Section: Theorem 63 Consider T a Continuous Linear Operatormentioning

confidence: 80%

“…and a suitable sequence h ∈ q v s ( ). The characterization for pseudodifferential operators in Theorem 3.2 [2] says that P is a Weyl operator P = Op w (σ ) with a symbol σ in M ∞,q 1⊗v s (R 2d ). Since T and therefore T * are invertible on L 2 (R d ), P is also invertible on L 2 (R d ).…”

Section: Theorem 63 Consider T a Continuous Linear Operatormentioning

confidence: 99%

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Wigner Analysis of Operators. Part II: Schrödinger Equations

Cordero,

Giacchi,

Rodino

2024

Commun. Math. Phys.

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We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schrödinger equations with bounded perturbations. To reach this goal, we perform a so-called $${\mathcal {A}}$$ A -Wigner analysis of the previous equations, as started in Part I, cf. Cordero and Rodino (Appl Comput Harmon Anal 58:85–123, 2022). Namely, the classical Wigner distribution is extended by considering a class of time–frequency representations constructed as images of metaplectic operators acting on symplectic matrices $${\mathcal {A}}\in Sp(2d,\mathbb {R})$$ A ∈ S p ( 2 d , R ) . Sub-classes of these representations, related to covariant symplectic matrices, reveal to be particularly suited for the time–frequency study of the Schrödinger evolution. This testifies the effectiveness of this approach for such equations, highlighted by the development of a related wave front set. We first study the properties of $${\mathcal {A}}$$ A -Wigner representations and related pseudodifferential operators needed for our goal. This approach paves the way to new quantization procedures. As a byproduct, we introduce new quasi-algebras of generalized metaplectic operators containing Schrödinger equations with more general potentials, extending the results contained in the previous works (Cordero et al. in J Math Pures Appl 99(2):219–233, 2013, J Math Phys 55(8):081506, 2014).

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Section: Theorem 63 Consider T a Continuous Linear Operatormentioning

confidence: 73%

Section: Theorem 63 Consider T a Continuous Linear Operatormentioning

confidence: 80%

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Wigner Analysis of Operators. Part II: Schrödinger Equations

Cordero,

Giacchi,

Rodino

2024

Commun. Math. Phys.

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“…where, if we write w = (y, η), then χ(y, η) = (y + τ η, η). This striking result is due to the peculiar action of W on the phase Φ(x, ξ) = xξ − τ ξ 2 . It generalizes to quadratic Φ(x, ξ), corresponding to quadratic Hamiltonians and linear symplectic map χ in (4), see for example [20].…”

Section: Introductionmentioning

confidence: 99%

“…where z := (1 + |z| 2 ) 1/2 , in the spirit of the estimates for Gabor kernels, which have been widely investigated in the literature, classical references are [2,10,15,16,26,27], see also [18,Chapter 5].…”

Section: Introductionmentioning

confidence: 99%

Wigner analysis of fourier integral operators with symbols in the Shubin classes

Cordero,

Giacchi,

Rodino

et al. 2024

Nonlinear Differ. Equ. Appl.

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We study the decay properties of Wigner kernels for Fourier integral operators of types I and II. The symbol spaces that allow a nice decay of these kernels are the Shubin classes $$\Gamma ^m({\mathbb {R}^{2d}})$$ Γ m ( R 2 d ) , with negative order m. The phases considered are the so-called tame ones, which appear in the Schrödinger propagators. The related canonical transformations are allowed to be nonlinear. It is the nonlinearity of these transformations that are the main obstacles for nice kernel localizations when symbols are taken in the Hörmander’s class $$S^{0}_{0,0}({\mathbb {R}^{2d}})$$ S 0 , 0 0 ( R 2 d ) . Here we prove that Shubin classes overcome this problem and allow a nice kernel localization, which improves with the decreasing of the order m.

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Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

Cordero

Giacchi

2022

J. Pseudo-Differ. Oper. Appl.

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Characterization of Smooth Symbol Classes by Gabor Matrix Decay

Cited by 6 publications

References 39 publications

Wigner Analysis of Operators. Part II: Schrödinger Equations

Wigner Analysis of Operators. Part II: Schrödinger Equations

Wigner analysis of fourier integral operators with symbols in the Shubin classes

Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

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