arXiv subjects

Loredana Caso

Publications and source records attributed to Loredana Caso.

5 recordsLinked to original sources

arXiv2022

General Kernel estimates of Schrödinger type operators with unbounded diffusion terms

We prove first that the realization $A_{\min}$ of $A:=\mathrm{div}(Q\nabla)-V$ in $L^2(\mathbb{R}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2(\mathbb{R}^d)$ which coincides on $L^2(\mathbb{R}^d)\cap C_b(\mathbb{R}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b(\mathbb{R}^d)$. Moreover, using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min}$ in the case of polynomially and exponentially diffusion and potential coefficients.

math.AP↗

arXiv2021

Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^α)Δ+ b|x|^{α-2}x\cdot\nabla - c |x|^{α-2} - |x|^β , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $α\in[0,2)$, $β>0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$.

math.AP↗

arXiv2020

Gravitational Waves in Higher Order Teleparallel Gravity

The teleparallel equivalent of higher order Lagrangians like $L_{\Box R}=-R+a_{0}R^{2}+a_{1}R\Box R$ can be obtained by means of the boundary term $B=2\nabla_μT^μ$. In this perspective, we derive the field equations in presence of matter for higher-order teleparallel gravity considering, in particular, sixth-order theories where the $\Box$ operator is linearly included. In the weak field approximation, gravitational wave solutions for these theories are derived. Three states of polarization are found: the two standard $+$ and $\times$ polarizations, namely 2-helicity massless transverse tensor polarizations, and a 0-helicity massive, with partly transverse and partly longitudinal scalar polarization. Moreover, these gravitational waves exhibit four oscillation modes related to four degrees of freedom: the two classical $+$ and $\times$ tensor modes of frequency $ω_{1}$, related to the standard Einstein waves with $k^{2}_{1}=0$; two mixed longitudinal-transverse scalar modes for each frequencies $ω_{2}$ and $ω_{3}$, related to two different 4-wave vectors, $k^{2}_{2}=M_{2}^{2}$ and $k^{2}_{3}=M^{2}_{3}$. The four degrees of freedom are the amplitudes of each individual mode, i.e. $\hatε^{(+)}\left(ω_{1}\right)$, $\hatε^{(\times)}\left(ω_{1}\right)$, $\hat{B}_{2}\left(\mathbf{k}\right)$, and $\hat{B}_{3}\left(\mathbf{k}\right)$.

gr-qc↗

arXiv2020

Weak Field Limit and Gravitational Waves in $f(T,B)$ Teleparallel Gravity

We derive the gravitational waves for $f\left(T, B\right)$ gravity, an extension of teleparallel gravity containing the torsion scalar $T$ and the boundary term $B$, and demonstrate that it is equivalent to $f(R)$ gravity. Gravitational waves, in this theory, show three polarizations: the two standard, plus and cross, of General Relativity, which are purely transverse, massless tensor polarization modes with two-helicity, and an additional massive scalar mode with zero-helicity. The last one is a mix of a longitudinal mode and a transverse breathing mode. The boundary term $B$ excites the extra scalar polarization and the mass of scalar field breaks the symmetry of the TT gauge by adding a new degree of freedom, namely a single mixed scalar polarization.

gr-qc↗

arXiv2019

Weak field limit and gravitational waves in higher-order gravity

We derive the weak field limit for a gravitational Lagrangian density $L_{g}=(R+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$ where higher-order derivative terms in the Ricci scalar $R$ are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of $R\Box^{k}R$ terms. By using a suitable set of coefficients $a_{k}$, it is possible to find up to $(p+2)$ normal modes of oscillation with six polarization states with helicity 0 or 2. Here $p$ is the higher order term in the $\Box$ operator appearing in the gravitational Lagrangian. More specifically: the mode $ω_{1}$, with $k^{2}=0$, has transverse polarizations $ε_{μν}^{\left(+\right)}$ and $ε_{μν}^{\left(\times\right)}$ with helicity 2; the $(p+1)$ modes $ω_{m}$, with $k^{2}\neq0$, have transverse polarizations $ε_{μν}^{\left(1\right)}$ and non-transverse ones $ε_{μν}^{\left(\text{TT}\right)}$, $ε_{μν}^{\left(\text{TS}\right)}$, $ε_{μν}^{\left(L\right)}$ with helicity 0.

gr-qc↗