Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

Pravda-Starov, Karel; Rodino, Luigi; Wahlberg, Patrik

doi:10.1002/mana.201600410

Cited by 43 publications

(73 citation statements)

References 29 publications

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“…This sharpens the result of Theorem 1.6 and extends the one obtained in [25] in the autonomous case.…”

Section: 42supporting

confidence: 85%

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

Pravda-Starov¹

2018

Math. Ann.

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We study evolution equations associated to time-dependent dissipative nonselfadjoint quadratic operators. We prove that the solution operators to these nonautonomous evolution equations are given by Fourier integral operators whose kernels are Gaussian tempered distributions associated to non-negative complex symplectic linear transformations, and we derive a generalized Mehler formula for their Weyl symbols. Some applications to the study of the propagation of Gabor singularities (characterizing the lack of Schwartz regularity) for the solutions to non-autonomous quadratic evolution equations are given.2010 Mathematics Subject Classification. 35S05, 47D06.

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“…This sharpens the result of Theorem 1.6 and extends the one obtained in [25] in the autonomous case.…”

Section: 42supporting

confidence: 85%

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

Pravda-Starov¹

2018

Math. Ann.

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“…where the propagator e −tq w (x,D) is defined in terms of semigroup theory [9,19]. According to [15,Theorem 6.2] the Gabor wave front set propagates as stated in the following result. Let q be a quadratic form on T * R d defined by a symmetric matrix Q ∈ C 2d×2d , Re Q 0 and F = J Q.…”

Section: Propagation Of Singularities For Schrödinger Equationsmentioning

confidence: 99%

“…Under the additional assumption on the Poisson bracket {q, q} = 0, [15,Corollary 6.3] says that S = Ker(Re F ) and hence…”

Section: Propagation Of Singularities For Schrödinger Equationsmentioning

confidence: 99%

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The Gabor Wave Front Set of Compactly Supported Distributions

Wahlberg

2020

Applied and Numerical Harmonic Analysis

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We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set.Dedicated to Luigi Rodino on the occasion of his 70 th birthday IntroductionThe Gabor wave front set of tempered distributions was introduced by Hörmander 1991 [8]. The idea was to measure singularities of tempered distributions both in terms of smoothness and decay at infinity comprehensively. Using the short-time Fourier transform the Gabor wave front set can be described as the directions in phase spaceThe Gabor wave front set behaves differently than the classical C ∞ wave front set, also introduced by Hörmander 1971 (see [7, Chapter 8]). It is for example translation invariant. The Gabor wave front set is adapted to the Shubin calculus of pseudodifferential operators [16] where symbols have isotropic behavior in phase space. With respect to this calculus, the corresponding notion of characteristic set, and the Gabor wave front set, pseudodifferential operators are microlocal and microelliptic, similar to pseudodifferential operators with Hörmander symbols in their natural context.The main result of this note concerns the Gabor wave front set of compactly supported distributions. We show (see Corollary 3.4) that for u ∈ E ′ (R d ) we havewhere WF G denotes the Gabor wave front set, WF denotes the classical wave front set, and π 2 is the projection on the covariable (second) phase space R d coordinate. By [7, Theorem 8.1.3], π 2 WF(u) = Σ(u). The symbol Σ(u) denotes the cone complement of the space of directions in R d in an open conic neighborhood of which the Fourier transform of u ∈ E ′ (R d ) decays rapidly. The equality (1.1) thus describes exactly the Gabor wave front set of u ∈ E ′ (R d ) in terms of known ingredients in terms of WF(u): The space coordinate is zero and the frequency directions are exactly the "irregular" frequency directions of u.In the literature there are several concepts of global wave front sets apart from the Gabor wave front set. There is a parametrized version [17] and an Gelfand-Shilov version [2] of the same idea. Melrose [12] introduced the scattering wave front set which 2010 Mathematics Subject Classification. 35A18, 35A21, 46F05, 46F12.

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“…where Y ⊆ R d is a linear subspace and A ∈ M d×d (R) is a symmetric matrix that leaves Y invariant, see [23]. It then automatically leaves Y ⊥ invariant so can be written…”

Section: Singularities Of the Solutionsmentioning

confidence: 99%

Shubin type Fourier integral operators and evolution equations

Cappiello

Schulz

Wahlberg

2019

J. Pseudo-Differ. Oper. Appl.

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We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin's class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.2010 Mathematics Subject Classification. Primary: 53S30. Secondary: 35S10, 35A22. Key words and phrases. Fourier integral operator, Schrödinger equation, semigroup, perturbation. R. Schulz gratefully acknowledges support of the project "Fourier Integral Operators, symplectic geometry and analysis on noncompact manifolds" received by the University of Turin in form of an "I@Unito" fellowship as well as institutional support by the University of Hannover.

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Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

Cited by 43 publications

References 29 publications

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

The Gabor Wave Front Set of Compactly Supported Distributions

Shubin type Fourier integral operators and evolution equations

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