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Andrea Posilicano

Publications and source records attributed to Andrea Posilicano.

At least 19 recordsLinked to original sources
arXiv2026

The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions

We study the self-adjoint Hamiltonian that models the quantum dynamics of a one-dimensional (1D) three-body system consisting of a light particle interacting with two heavy ones through a zero-range force. For an attractive interaction we determine the behavior of the eigenvalues below the essential spectrum in the regime $\varepsilon\ll 1$, where $\varepsilon$ is proportional to the square root of the mass ratio. We show that the $n$-th eigenvalue behaves as $E_{n}(\varepsilon)=-α^{2}+|σ_{n}|α^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $α$ is a negative constant that explicitly relates to the physical parameters and $σ_{n}$ is either the $n$-th extremum or the $n$-th zero of the Airy function Ai, depending on the kind (respectively, bosons or fermions) of the two heavy particles. Additionally, we prove that the essential spectrum coincides with the half-line $[-\frac{α^2}{4+\varepsilon^{2}},+\infty)$.

math-ph↗
arXiv2026

Resolvent, spectrum and resonances for the acoustic operator with piecewise constant coefficients

We study the acoustic operator $A_{v,ρ}:=v^{2}ρ\nabla\!\cdotρ^{-1}\nabla$ with transmission conditions at the boundary of $Ω=Ω_{1}\cup\dots\cupΩ_{n}$, where the $Ω_{\ell}$'s are connected disjoint open bounded Lipschitz domains, the positive functions $v$ and $ρ$ are constant on each connected component of $Ω$ and $v=ρ=1$ on ${\mathbb R}^{3}\backslash\overlineΩ$. Through a formula for the resolvents difference $(-A_{v,ρ}+z)^{-1}-(-Δ+z)^{-1}$, we provide a Limiting Absorption Principle, determine the spectrum, which turns out to be purely absolutely continuous, and, in the case the connected components of $Ω$ are of class ${\mathcal C}^{1,α}$, characterize the resonance set. The second part of the paper is devoted to the case where $Ω=Ω(\varepsilon)$ is connected with a small size $\varepsilon$ and the $\varepsilon$-analytic functions $v=v(\varepsilon)$ and/or $ρ=ρ(\varepsilon)$ converge to $0_{+}$ inside $Ω(\varepsilon)$ as $\varepsilon\downarrow 0$; there, we provide the analytic $\varepsilon$-expansions of the resonances of $A_{v,ρ}$ according to different choices of the rate of convergence towards zero of the material parameters.

math.AP↗
arXiv2024

The point scatterer approximation for wave dynamics

Given an open, bounded and connected set $Ω\subset\mathbb{R}^{3}$ and its rescaling $Ω_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$ (\varepsilon^{-2}χ_{Ω_{\varepsilon}}+χ_{\mathbb{R}^{3}\backslashΩ_{\varepsilon}})\partial_{tt}u=Δu+f $$ with initial data and source supported outside $Ω_{\varepsilon}$; here, $χ_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^τ),L^{2}(\mathbb{R}^{3})) $-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(Ω)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.

math-ph↗
arXiv2024

Body resonances for classical waves

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}χ_{Ω_{\varepsilon}}+χ_{Ω^{c}_{\varepsilon}})Δ$ modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the dynamics of acoustic waves in the presence of a small inhomogeneity of size $\varepsilon$ having high contrast $\varepsilon^{-2}$. In particular, we give precise results on the localization of the resonances of $H(\varepsilon)$ and their first-order $\varepsilon$-expansions; the latter are explicitly expressed in terms of the eigenvalues and eigenvectors of the Newton potential operator of the set $Ω$ whose rescaling of size $\varepsilon$ defines $Ω_{\varepsilon}$.

math.SP↗
arXiv2024

On the resolvent of $H+A^{*}+A$

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of $H+A^{*}+A$, Math. Phys. Anal. Geom. (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind $H+A^{*}+A$, where $H$ and $A$ play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Krein-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind $H+A^{*}_{n}+A_{n}-E_{n}$, the bounded operator $E_{n}$ playing the role of a renormalizing counter term.

math-ph↗
arXiv2023

Scattering theory with both regular and singular perturbations

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(Δ_B,Δ)$, where $Δ$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $Δ_B$ is a self-adjoint operator in a class of Laplacians with both a regular perturbation, given by a short-range potential, and a singular one describing boundary conditions (like Dirichlet, Neumann and semi-transparent $δ$ and $δ'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations.

math-ph↗
arXiv2023

Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetry. We show that a perturbation which breaks the $\mathbf{SU}(1,1)$-symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a perturbation which breaks both the $\mathbf{SU}(1,1)$-symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. The developing instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $\pm 2ω\mathrm{i}$.

math.AP↗
arXiv2022

Inverse wave scattering in the time domain for point scatterers

Let $Δ_{α,Y}$ be the bounded from above self-adjoint realization in $L^{2}({\mathbb R}^{3})$ of the Laplacian with $n$ point scatterers placed at $Y=\{y_{1},\dots,y_{n}\}\subset{\mathbb R}^{3}$, the parameters $(α_{1},\dotsα_{n})\equivα\in {\mathbb R}^{n}$ being related to the scattering properties of the obstacles. Let $u^{α,Y}_{f_ε}$ and $u^{\varnothing}_{f_ε}$ denote the solutions of the wave equations corresponding to $Δ_{α,Y}$ and to the free Laplacian $Δ$ respectively, with a source term given by the pulse $f_ε(x)=\sum_{k=1}^{N}f_{k}\,φ_ε(x-x_{k}) $ supported in $ε$-neighborhoods of the points in $X_{N}=\{x_{1},\dots, x_{N}\}$, $X_{N}\cap Y=\varnothing$. We show that, for any fixed $λ>\supσ(Δ_{α,Y})$, there exits $N_{\circ}\ge 1$ such that the locations of the points in $Y$ can be determined by the knowledge of the finite-dimensional scattering data operator $F^{N}_λ:{\mathbb R}^{N}\to{\mathbb R}^{N}$, $N\ge N_{\circ}$, $$ (F^{N}_λf)_{k}:=\lim_{ε\searrow 0}\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^{α,Y}_{f_ε}(t,x_{k})-u^{\varnothing}_{f_ε}(t,x_{k})\big)\,dt\,. $$ We exploit the factorized form of the resolvent difference $(-Δ_{α,Y}+λ)^{-1}-(-Δ+λ)^{-1}$ and a variation on the finite-dimensional factorization in the MUSIC algorithm; multiple scattering effects are not neglected.

math-ph↗
arXiv2022

On the Origin of Minnaert Resonances

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size $\varepsilon$), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency $ω_{M}$. Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency $ω$ is described by a self-adjoint $ω$-dependent Schrödinger operator with a singular $δ$-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to $\varepsilon\ll1$) such an operator has a non-trivial limit (i.e., it asymptotically differs from the Laplacian) if and only if $ω=ω_{M}$. The limit operator describing the non-trivial scattering process is explicitly determined and belongs to the class of point perturbations of the Laplacian. When the frequency of the incident wave approaches $ω_{M}$, the scattering process undergoes a transition between an asymptotically trivial behaviour and a non-trivial one.

math.AP↗
arXiv2020

The semi-classical limit with a delta-prime potential

We consider the quantum evolution $e^{-i\frac{t}{\hbar}H_β} ψ_ξ^{\hbar}$ of a Gaussian coherent state $ψ_ξ^{\hbar}\in L^{2}(\mathbb{R})$ localized close to the classical state $ξ\equiv (q,p) \in \mathbb{R}^{2}$, where $H_β$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}} + β\,δ'_{0}$, with $δ'_{0}$ the derivative of Dirac's delta distribution at $x = 0$ and $β$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(\mathbb{R})$-norm, uniformly for any $t \in \mathbb{R}$ away from the collision time) by $e^{\frac{i}{\hbar} A_{t}} e^{it L_{B}} ϕ^{\hbar}_{x}$, where $A_{t} = \frac{p^{2}t}{2m}$, $ϕ_{x}^{\hbar}(ξ) := ψ^{\hbar}_ξ(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $\mathcal{C}^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0} := \{(q,p) \in \mathbb{R}^{2}\,|\,q \neq 0\}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\hbar^{7/2-λ}$, $0 < λ< 1/2$, whereas it turns out to be of order $\hbar^{3/2-λ}$, $0 < λ< 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

math-ph↗
arXiv2020

On the self-adjointness of H+A*+A

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\hat H$ of the formal Hamiltonian $H+A^{*}+A$ with $D(H)\cap D(\hat H)=\{0\}$. We give an explicit characterization of $D(\hat H)$ and provide a formula for the resolvent difference $(-\hat H+z)^{-1}-(-H+z)^{-1}$. Moreover, we consider the problem of the description of $\hat H$ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$, where the $A_{n}\!$'s are regularized operators approximating $A$ and the $E_{n}$'s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Krein's resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.

math-ph↗
arXiv2020

Inverse wave scattering in the Laplace domain: a factorization method approach

Let $Δ_Λ\le λ_Λ$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle $Ω$. Let $u^Λ_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $Δ_Λ$ and to the free Laplacian $Δ$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $λ>λ_Λ\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\barΩ$, the obstacle $Ω$ can be reconstructed by the data $$ F^Λ_λf(x):=\int_{0}^{\infty}e^{-\sqrtλ\,t}\big(u^Λ_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,. $$ A similar result holds in the case of screens reconstruction, when the boundary conditions are assigned only on a part of the boundary. Our method exploits the factorized form of the resolvent difference $(-Δ_Λ+λ)^{-1}-(-Δ+λ)^{-1}$.

math.AP↗
arXiv2020

The semiclassical limit on a star-graph with Kirchhoff conditions

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 Δ$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\uın's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\oplus})$, where $H_{D}^{\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.

math-ph↗
arXiv2020

Scattering from local deformations of a semitransparent plane

We study scattering for the couple $(A_{F},A_{0})$ of Schrödinger operators in $L^2(\mathbb{R}^3)$ formally defined as $A_0 = -Δ+ α\, δ_{π_0}$ and $A_F = -Δ+ α\, δ_{π_F}$, $α>0$, where $δ_{π_F}$ is the Dirac $δ$-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function $F:\mathbb{R}^{2}\to\mathbb{R}$ and $π_{0}$ is the undeformed plane corresponding to the choice $F\equiv 0$. We provide a Limiting Absorption Principle, show asymptotic completeness of the wave operators and give a representation formula for the corresponding Scattering Matrix $S_{F}(λ)$. Moreover we show that, as $F\to 0$, $\|S_{F}(λ)-\mathsf 1\|^{2}_{\mathfrak{B}(L^{2}({\mathbb S}^{2}))}={\mathcal O}\!\left(\int_{\mathbb{R}^{2}}d\textbf{x}|F(\textbf{x})|^γ\right)$, $0<γ<1$. We correct a minor mistake in the computation of the scattering matrix, occurring in the published version of this paper (see J. Math. Anal. Appl. 473(1) (2019), pp. 215-257). The mistake was in Section 7, and affected the statement of Corollary 7.2, specifically, Eq. (7.8). Regrettably the formula for $S_F$ in the Corrigendum J. Math. Anal. Appl. 482(1) (2020), 123554, still contains a misprint, the correct expression is the one given here.

math-ph↗
arXiv2019

Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\widetildeΔ,Δ)$, where $Δ$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetildeΔ$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetildeΔ)\, :\, Δu=\widetildeΔu\}$ is dense. Typically $\widetildeΔ$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at $Γ=\partialΩ$, where $Ω\subset{\mathbb R}^{3}$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $Σ\subsetΓ$, a relatively open subset with a Lipschitz boundary. We show that either $Γ$ or $Σ$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.

math.AP↗
arXiv2019

The semi-classical limit with delta potentials

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian $\mathsf H$ is given, as sum of quadratic forms, by $\mathsf H= -\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,αδ_{0}$, with $α\in\mathbb R$ and $δ_{0}$ the Dirac delta-distribution at $x=0$. We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order $\hbar^{3/2-λ}$, $0\!<\!λ\!<\!3/2$, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to $\mathcal C^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0}:=\{(q,p)\!\in\!\mathbb R^{2}\,|\,q\!\not=\!0\}$, of ($-i$ times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.

math-ph↗
arXiv2019

Asymptotic Completeness and S-Matrix for Singular Perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

math-ph↗