Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials

We consider the $0$-order perturbed Lamé operator $-\Delta^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-\Delta^\ast$ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about $V$

The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations

We prove an $L^p$-version of the limiting absoprtion principle for a class of periodic elliptic differential operators of second order. The result is applied to the construction of nontrivial solutions of nonlinear Helmholtz equations with periodic coefficient functions

Scattering in the energy space for the NLS with variable coefficients

We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\simeq|u|^{\gamma-1}u$. We assume that $L$ is a small, long range perturbation of $\Delta$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow $e^{itL}$, we prove global well posedness in the energy space for subcritical powers $\gamma1+\frac4n$. When the domain is $\mathbb{R}^{n}$, by extending the Strichartz estimates due to Tataru [Tataru08], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

In this work we investigate the L^p-L^q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an L^p-L^q-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell’s equations

In this work we investigate the $L^p-L^q$-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an $L^p-L^q$-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems

Global Strichartz estimates for an inhomogeneous Maxwell system

We show a global in time Strichartz estimate for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as $|x|\rightarrow\infty$ and a non-trapping condition. The proof is based on smoothing estmates in weighted $L^2$ spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem

Helmholtz and dispersive equations with variable coefficients on exterior domains

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equation \begin{equation*} -Lu+zu=f, \qquad -Lu:=\nabla^{b}(a(x)\nabla^{b}u)-c(x)u, \qquad \nabla^{b}:=\nabla+ib(x) \end{equation*} with fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping. As an application, we prove smoothing estimates for the Schr\"{o}dinger flow $e^{itL}$ and the wave flow $e^{it \sqrt{L}}$ with variable coefficients on exterior domains and Dirichlet boundary conditions