arXiv subjects

Anna Rita Giammetta

Publications and source records attributed to Anna Rita Giammetta.

3 recordsLinked to original sources

arXiv2016

Hardy inequality and fractional Leibnitz rule for perturbed Hamiltonians on the line

We consider the following perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 + V(x)$ on the real line. The potential $V(x)$ is a real - valued function of short range type. We study the equivalence of classical homogeneous Sobolev type spaces $\dot{H}^s_p$, $p \in (1,\infty)$ and the corresponding perturbed homogeneous Sobolev spaces associated with the perturbed Hamiltonian. It is shown that the assumption zero is not a resonance guarantees that the perturbed and unperturbed homogeneous Sobolev norms of order $s = γ- 1 \in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Sobolev spaces of order $s \in [0,1/p)$ invariant.

math.AP↗

arXiv2016

On homogeneous Besov spaces for $1D$ Hamiltonians without zero resonance

We consider 1-D Laplace operator with short range potential V(x), such that $$(1+|x|)^γV(x) \in L^1(R), \ \ γ> 1.$$ We study the equivalence of classical homogeneous Besov type spaces $\dot{B}^s_p(R)$, $p \in (1,\infty)$ and the corresponding perturbed homogeneous Besov spaces associated with the perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 + V(x)$ on the real line. It is shown that the assumptions $1/p < γ-1$ and zero is not a resonance guarantee that the perturbed and unperturbed homogeneous Besov norms of order $s \in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Besov spaces of order $s \in [0,1/p)$ invariant.

math.AP↗

arXiv2014

Fractional Schrödinger-Poisson-Slater system in one dimension

In this paper we study local and global well-posedness of the following Cauchy problem: $$ \bigg \{ \begin{array}{rl} i\partial_tΨ+\frac{1}{2}Δ_{x}Ψ= A_0Ψ+α|Ψ|^{γ-1}Ψ, & (t,x)\in\R\times\R,\\ (-Δ_{x})^{σ/2}A_0= |Ψ|^2, & \\ Ψ(0,\cdot)=f, & \end{array} $$ with $σ\in(0,1)$, $α=\pm 1$, $1<γ\leq 5$, in the spaces $L^2(\mathbf{R})$ and $H^1(\mathbf{R})$.

math.AP↗