Searcharxiv⌕ Search

arXiv subjects

Elena Cordero

Publications and source records attributed to Elena Cordero.

At least 73 records · Page 4Linked to original sources
arXiv2015

Generalized Metaplectic Operators and the Schrödinger Equation with a Potential in the Sjöstrand Class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential opertators in a Sjöstrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phase-space concentration, as measured by modulation space norms.

math.AP↗
arXiv2015

Schrödinger equations with rough Hamiltonians

We consider a class of linear Schrödinger equations in R^d with rough Hamiltonian, namely with certain derivatives in the Sjöstrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class. Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.

math.AP↗
arXiv2015

Gabor Frames of Gaussian Beams for the Schrödinger equation

The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of $\hbar$-Gabor frames, as recently defined by M. de Gosson, and we provide precise $L^2$-estimates of their accuracy, in terms of the Planck constant $\hbar$. Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.

math.AP↗
arXiv2014

Gabor wave packets and evolution operators

We perform a Gabor analysis for a large class of evolution equations with constant coefficients. We show that the corresponding propagators have a very sparse Gabor matrix, displaying off-diagonal exponential decay. The results apply to hyperbolic, weakly hyperbolic and parabolic equations. Some numerical experiments are provided.

math.FA↗
arXiv2014

On the Schrödinger equation with potential in modulation spaces

This work deals with Schrödinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such evolution as the composition of a metaplectic operator and a pseudodifferential operator having symbol in certain classes of modulation spaces. About propagation of singularities, we use a new notion of wave front set, which allows the expression of optimal results of propagation in our context. To support this claim, many comparisons with the existing literature are performed in this work.

math.AP↗
arXiv2014

Triangular Subgroups of $Sp(d,{\mathbb R})$ and Reproducing Formulae

We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ between the Heisenberg group and the symplectic group. Subgroups $H=Σ\rtimes D$, with $Σ$ being a $d\times d$ symmetric matrix and $D$ a closed subgroup of $GL(d,{\mathbb R})$, are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in [5]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrödinger representation of ${\mathbb R}^{2d}$ or the wavelet representation of ${\mathbb R}^d\rtimes D$, with $D$ closed subgroup of $GL(d,{\mathbb R})$. Finally, we shall provide new examples of reproducing groups of the type $H=Σ\rtimes D$, in dimension $d=2$.

math.RT↗
arXiv2013

Schrödinger type propagators, pseudodifferential operators and modulation spaces

We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of nondegenerate generalized quadratic forms that includes Schrödinger propagators and pseudodifferential operators. As a byproduct we obtain a characterization of all exponents $p,q,r_1,r_2,t_1,t_2 \in [1,\infty]$ of modulation spaces such that a symbol in $M^{p,q}(\mathbb R^{2d})$ gives a pseudodifferential operator that is continuous from $M^{r_1,r_2}(\mathbb R^d)$ into $M^{t_1,t_2}(\mathbb R^d)$.

math.FA↗
arXiv2012

The Wiener Property for a Class of Fourier Integral Operators

We construct a one-parameter family of algebras consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of this class of operators. The operator algebra is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.

math.FA↗
arXiv2011

Approximation of Fourier Integral Operators by Gabor multipliers

A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis.

math.FA↗
arXiv2011

The Cauchy Problem for the Vibrating Plate Equation in modulation spaces

The local solvability of the Cauchy problem for the nonlinear vibrating plate equation is showed in the framework of modulation spaces. In the opposite direction, it is proved that there is no local wellposedness in Wiener amalgam spaces even for the solution to the homogeneous vibrating plate equation.

math.AP↗
arXiv2011

Strichartz Estimates for the Vibrating Plate Equation

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schrödinger-type equations we show its close relation with the Schrödinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.

math.AP↗
arXiv2010

Dilation properties for weighted modulation spaces

In this paper we give a sharp estimate on the norm of the scaling operator $U_λf(x)=f(λx)$ acting on the weighted modulation spaces $\M{p,q}{s,t}(\R^{d})$. In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well posedeness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

math.FA↗
arXiv2009

Boundedness of Schroedinger type propagators on modulation spaces

It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces $\cM^{p,q}$, for $1\leq p=q\leq\infty$, provided their symbols belong to the Sjöstrand class $M^{\infty,1}$. However, they generally fail to be bounded on $\cM^{p,q}$ for $p\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\cM^{p,q}$ for $p\not=q$, and between $\cM^{p,q}\to\cM^{q,p}$, $1\leq q< p\leq\infty$. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.

math.FA↗
arXiv2009

Sharp Continuity Results for the Short-Time Fourier Transform and for Localization Operators

We completely characterize the boundedness on $L^p$ spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on $L^p$ spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces $W(L^p,L^q)$ are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for localization operators on $L^p$ spaces and prove the optimality of our results. More generally, we prove sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.

math.AP↗
arXiv2009

Pseudodifferential operators on $L^p$, Wiener amalgam and modulation spaces

We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.

math.AP↗
arXiv2008

Boundedness of Fourier Integral Operators on $\mathcal{F} L^p$ spaces

We study the action of Fourier Integral Operators (FIOs) of H{ö}rmander's type on ${\mathcal{F}} L^p({\mathbb {R}}^d_{comp}$, $1\leq p\leq\infty$. We see, from the Beurling-Helson theorem, that generally FIOs of order zero fail to be bounded on these spaces when $p\not=2$, the counterexample being given by any smooth non-linear change of variable. Here we show that FIOs of order $m=-d|1/2-1/p|$ are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension $d\geq1$, even for phases which are linear in the dual variables. The proofs make use of tools from time-frequency analysis such as the theory of modulation spaces.

math.AP↗
arXiv2008

On the global boundedness of Fourier integral operators

We consider a class of Fourier integral operators, globally defined on $\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented.

math.FA↗