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Luca Fanelli

Publications and source records attributed to Luca Fanelli.

At least 55 records · Page 3Linked to original sources
arXiv2017

Gaussian decay of Harmonic Oscillators and related models

We prove that the decay of the eigenfunctions of harmonic oscillators, uniform electric or magnetic fields is not stable under 0-order complex perturbations, even if bounded, of these Hamiltonians, in the sense that we can produce solutions to the evolutionary Schrödinger flows associated to the Hamiltonians, with a stronger Gaussian decay at two distinct times. We then characterize, in a quantitative way, the sharpest possible Gaussian decay of solutions as a function of the oscillation frequency or the strength of the field, depending on the Hamiltonian which is considered. This is connected to the Hardy's Uncertainty Principle for free Schrödinger evolutions.

math.AP↗
arXiv2016

Spectral stability of Schroedinger operators with subordinated complex potentials

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for Schroedinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities.

math.SP↗
arXiv2016

Spherical Schrödinger Hamiltonians: Spectral Analysis and Time Decay

In this survey, we review recent results concerning the canonical dispersive flow $e^{itH}$ led by a Schrödinger Hamiltonian $H$. We study, in particular, how the time decay of space $L^p$-norms depends on the frequency localization of the initial datum with respect to the some suitable spherical expansion. A quite complete description of the phenomenon is given in terms of the eigenvalues and eigenfunctions of the restriction of $H$ to the unit sphere, and a comparison with some uncertainty inequality is presented.

math-ph↗
arXiv2015

Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows

We consider a Schrödinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then $ \||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.

math.AP↗
arXiv2013

Relativistic Hardy inequalities in magnetic fields

We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best constant as in the free case. This leaves naturally open an interesting question whether there exist magnetic fields for which a Hardy inequality with a better constant than the usual one, in connection with the well known diamagnetic phenomenon arising in non-relativistic models.

math-ph↗
arXiv2013

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions

We prove a sharp version of the Hardy uncertainty principle for Schrödinger equations with external bounded electromagnetic potentials, based on logarithmic convexity properties of Schrödinger evolutions. We provide, in addition, an example of a real electromagnetic potential which produces the existence of solutions with critical gaussian decay, at two distinct times.

math.AP↗
arXiv2012

Hardy Uncertainty Principle and unique continuation properties of covariant Schrodinger flows

We prove a logarithmic convexity result for exponentially weighted $L^2$-norms of solutions to electromagnetic Schrödinger equation, without needing to assume smallness of the magnetic potential. As a consequence, we can prove a unique continuation result in the style of the Hardy uncertainty principle, which generalizes the analogous theorems which have been recently proved by Escauriaza, Kenig, Ponce and Vega.

math.AP↗
arXiv2012

Time decay of scaling critical electromagnetic Schrödinger flows

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp $L^{1}\to L^{\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov-Bohm potentials.

math.AP↗
arXiv2011

The lack of compactness in the Sobolev-Strichartz inequalities

We provide a general method to decompose any bounded sequence in $\dot H^s$ into linear dispersive profiles generated by an abstract propagator, with a rest which is small in the associated Strichartz norms. The argument is quite different from the one proposed by Bahouri-Gérard and Keraani in the cases of the wave and Schrödinger equations, and is adaptable to a large class of propagators, including those which are matrix-valued.

math.AP↗
arXiv2011

On the lack of dispersion for a class of magnetic Dirac flows

We show that global Strichartz estimates for magnetic Dirac operators generally fails, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, i. e. with homogeneity-degree more than -1, such that the magnetic field points to a fixed direction, which does not depend on $x\in \mathbb{R}^3$.

math.AP↗
arXiv2011

Carleman estimates and necessary conditions for the existence of waveguides

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the exponential decay in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $W$ at infinity.

math.AP↗
arXiv2011

A priori estimates for the Helmholtz equation with electromagnetic potentials in exterior domains

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension $n\geq3$. We prove, by multiplier techniques in the sense of Morawetz, a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to the absence of embedded eigenvalues and zero-resonances, under explicit conditions on the potentials.

math.AP↗
arXiv2009

Virial identity and weak dispersion for the magnetic Dirac equation

We analyze the dispersive properties of a Dirac system perturbed with a magnetic field. We prove a general virial identity; as applications, we obtain smoothing and endpoint Strichartz estimates which are optimal from the decay point of view. We also prove a Hardy-type inequality for the perturbed Dirac operator.

math.AP↗
arXiv2009

Counterexamples to Strichartz estimates for the magnetic Schroedinger equation

In space dimension $n\geq3$, we consider the magnetic Schrödinger Hamiltonian $H=-(\nabla-iA(x))^2$ and the corresponding Schrödinger equation i\partial_tu+Hu=0. We show some explicit examples of potentials $A$, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.

math.AP↗
arXiv2009

Endpoint Strichartz estimates for the magnetic Schrodinger equation

We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a non trapping condition, which are expressed as smallness of suitable components of the potentials. However, the potentials themselves can be large, and we avoid completely any a priori spectral assumption on the operator. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.

math.AP↗