arXiv subjects

Federica Gregorio

Publications and source records attributed to Federica Gregorio.

14 recordsLinked to original sources

arXiv2026

Higher order Schrödinger operators

In this paper we consider higher order Schrödinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients differential operator $L=Δ^2$. Subsequently, we extend our results to more general operators $L$ featuring suitable variable coefficients. We are interested in domain characterization and generation properties of these operators in $L^p(\mathbb{R}^N)$ for $p \in (1, \infty)$. To address this problems we employ a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss and we prove that the $L^p$-realization of $\mathcal L$ is quasi sectorial and, consequently, generates an analytic semigroup. Furthermore, this approach allows for a sharp characterization of the operator's domain as the intersection of the domains of the bilaplacian and the multiplication operator. The required assumptions allow to treat potentials that grow at infinity like $|x|^r$ for some $r<4$.

math.AP↗

arXiv2025

Fourth-order operators with unbounded coefficients in $L^1$ spaces

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterise the maximal domain of $A$ in $L^1(\mathbb{R}^N)$.

math.FA↗

arXiv2024

Controllability of Forward Stochastic Reaction--Convection--Diffusion Systems with Cascade Structure

We investigate the null and approximate controllability of coupled linear forward stochastic reaction--convection--diffusion systems under suitable cascade coupling conditions. The model consists of two forward stochastic parabolic equations governed by general second-order differential operators with time-, space-, and random-dependent coefficients. We consider a localized control acting on the drift term of the first equation together with controls on the diffusion terms. By a duality argument, the controllability problem is reduced to an observability problem for the associated adjoint backward stochastic parabolic system. The main contribution of this paper is the establishment of a new global Carleman estimate for coupled backward stochastic parabolic systems whose drift terms belong to a negative Sobolev space. This estimate yields the required observability properties and, consequently, the null and approximate controllability of the original system.

math.OC↗

arXiv2024

Fourth-order operators with unbounded coefficients

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with $|D a(x)|\leq c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterize the maximal domain of such operators in $L^p(\mathbb{R}^N)$ for $1<p<\infty$.

math.AP↗

arXiv2022

Fourth-order Schrödinger type operator with unbounded coefficients in $L^2(\mathbb{R}^N)$

In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schrödinger type operator with unbounded coefficients of the form $$A=a^{2} Δ^2+V^{2}$$ where $a(x)=1+|x|^α$ and $V=|x|^β$ with $α>0$ and $β>(α-2)^+$. We obtain that $(-A,D(A))$ generates an analytic strongly continuous semigroup in $L^2(\mathbb{R}^N)$ for $N\geq5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2α-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]

math.AP↗

arXiv2021

Bi-Kolmogorov type operators and weighted Rellich's inequalities

In this paper we consider the symmetric Kolmogorov operator $L=Δ+\frac{\nabla μ}μ\cdot \nabla$ on $L^2(\mathbb R^N,dμ)$, where $μ$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $μ$ we prove first weighted Rellich's inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,dμ)$ and $H^4(\mathbb R^N,dμ)$ respectively, generate analytic semigroups of contractions on $L^2(\mathbb R^N,dμ)$. We observe that $dμ$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2(\mathbb R^N,dμ)$.

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

Higher order operators on networks: hyperbolic and parabolic theory

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions.

math.AP↗

arXiv2020

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

We analyze properties of semigroups generated by Schrödinger operators $-Δ+V$ or polyharmonic operators $-(-Δ)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kre\uın-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

math.SP↗

arXiv2019

Bi-Laplacians on graphs and networks

We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete Laplacian acting on a connected discrete graph $\mathsf{G}$. For both operators we discuss well-posedness of the associated {linear} parabolic problems \[ \frac{\partial u}{\partial t}=-Au,\qquad\frac{df}{dt}=-\mathcal L^2 f, \] on $L^p(\mathcal{G})$ or $\ell^p(\mathsf{V})$, respectively, for $1\leq p\leq\infty$. In view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order $2N$ for $N>1$, our most surprising finding is that, after some transient time, the parabolic equations driven by $-A$ may display Markovian features, depending on the imposed transmission conditions in the vertices. Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by $-\mathcal L^2$ is also presented.

math.AP↗

arXiv2018

On Lennard-Jones-type potentials on the half-line

In this paper we study a particle under the influence of a Lennard-Jones potential moving in a simple quantum wire modelled by the positive half-line. Despite its physical significance, this potential is only rarely studied in the literature and due to its singularity at the origin it cannot be considered as a standard perturbation of the one-dimensional Laplacian. It is therefore our aim to provide a thorough description of the Hamiltonian in one dimension via the construction of a suitable quadratic form. Our results include a discussion of spectral and scattering properties which finally allows us to generalise some results from [Robinson1974] as well as [RadinSimon1978].

math-ph↗

arXiv2017

Weighted Hardy's inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $dμ$. Moreover, the optimality of the constant $c_{0,μ}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $dμ$ allow the drift term to be of type $\frac{\nabla μ}μ= -|x|^{m-2}x$ with $m> 0$.

math.AP↗

arXiv2016

Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$-spaces

In this paper we give sufficient conditions on $α\geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^α)Δu+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.

math.AP↗

arXiv2016

Fourth-order Schrödinger type operator with singular potentials

In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=Δ^2-V=Δ^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $ΔA^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.

math.AP↗