arXiv subjects
Publications and source records attributed to Fernando Cardoso.
We prove optimal high-frequency resolvent estimates for perturbations of the Laplacian by large long-range magnetic and electric potentials in all dimensions $n\ge 3$. As an application, we prove dispersive estimates for the corresponding wave group in the case $n=3$.
We prove dispersive estimates for the wave and Schrodinger groups associated to a second-order elliptic self-adjoint operator depending on a semi-classical parameter. Applications are made to non-trapping metric perturbations and to perturbations by a magnetic potential.
We prove optimal dispersive estimates for the wave group $e^{it\sqrt{-Δ+V}}$ for a class of real-valued potentials $V\in C^{(n-3)/2}({\bf R}^n)$, $4\le n\le 7$, such that $\partial_x^αV(x)=O(|x|^{-δ})$ for $|x|>1$, where $δ>(n+1)/2$.
We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(|x|^{-δ})$, $δ>n-1$, and $V\in C^k({\bf R}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)/2\le k_n<n/2$. We also give a sufficient condition in terms of $L^1\to L^\infty$ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.
We study the transmission problem in bounded domains with dissipative boundary conditions. Under some natural assumptions, we prove uniform bounds of the corresponding resolvents on the real axis at high frequency, and as a consequence, we obtain free of eigenvalues regions. As an application, we get exponential decay of the energy of the solutions of the correpsonding mixed boundary value problems.
We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-Δ+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.
We obtain large time decay estimates on weighted $L^p$ spaces for solutions to the wave equation with real-valued potential $V(x)=O(|x|^{-2-a})$, $a>0$, for $|x|>1$.
We show how certain hypergeometric functions play an important role in finding fundamental solutions for a generalized Tricomi operator.
Partial Fourier transforms are used to find explicit formulas for two remarkable fundamental solutions for a generalized Tricomi operator. These fundamental solutions reflect clearly the mixed type of the operator. In order to prove these results, we establish explicit formulas for Fourier transforms of some type of Bessel functions.