arXiv subjects

Claudio Cacciapuoti

Publications and source records attributed to Claudio Cacciapuoti.

At least 19 recordsLinked to original sources

arXiv2025

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 $E_{n}(\varepsilon)=-\alpha^{2}+|\sigma_{n}|\alpha^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $\alpha$ is a negative constant that explicitly relates to the physical parameters and $\sigma_{n}$ 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 $[-\frac{\alpha^2}{4+\varepsilon^{2}},+\infty)$.

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

Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan--Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.

math-ph↗

arXiv2021

Well posedness of the nonlinear Schrödinger equation with isolated singularities

We study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers.

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

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

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

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

We consider a compact metric graph of size $\varepsilon$, and attach to it several edges (leads) of length of order one (or of infinite length). As $\varepsilon$ goes to zero, the graph $\mathcal{G}^\varepsilon$ obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On $\mathcal{G}^\varepsilon$ we define an Hamiltonian $H^\varepsilon$, properly scaled with the parameter $\varepsilon$. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates $H^\varepsilon$ (in a suitable norm resolvent sense) as $\varepsilon\to0$. The effective Hamiltonian depends on the spectral properties of an auxiliary $\varepsilon$-independent Hamiltonian defined on the compact graph obtained by setting $\varepsilon = 1$. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit $\varepsilon\to0$, the leads are decoupled.

math-ph↗

arXiv2019

Existence of the ground state for the NLS with potential on graphs

We review and extend several recent results on the existence of the ground state for the nonlinear Schrödinger (NLS) equation on a metric graph. By ground state we mean a minimizer of the NLS energy functional constrained to the manifold of fixed $L^2$-norm. In the energy functional we allow for the presence of a potential term, of delta-interactions in the vertices of the graph, and of a power-type focusing nonlinear term. We discuss both subcritical and critical nonlinearity. Under general assumptions on the graph and the potential, we prove that a ground state exists for sufficiently small mass, whenever the constrained infimum of the quadratic part of the energy functional is strictly negative.

math-ph↗

arXiv2018

On inverses of Krein's Q-functions

Let $A_{Q}$ be the self-adjoint operator defined by the $Q$-function $Q:z\mapsto Q_{z}$ through the Krein-like resolvent formula $$(-A_{Q}+z)^{-1}= (-A_{0}+z)^{-1}+G_{z}WQ_{z}^{-1}VG_{\bar z}^{*}\,,\quad z\in Z_{Q}\,,$$ where $V$ and $W$ are bounded operators and $$Z_{Q}:=\{z\inρ(A_{0}):\text{$Q_{z}$ and $Q_{\bar z }$ have a bounded inverse}\}\,.$$ We show that $$Z_{Q}\not=\emptyset\quad\Longrightarrow\quad Z_{Q}=ρ(A_{0})\cap ρ(A_{Q})\,.$$ We do not suppose that $Q$ is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that $Q$ is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.

math.SP↗

arXiv2018

Variational and stability properties of constant solutions to the NLS equation on compact metric graphs

We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

math.AP↗

arXiv2018

The three-body problem in dimension one: From short-range to contact interactions

We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in norm resolvent sense. The two-body rescaled potentials are of the form $v^{\varepsilon}_σ(x_σ)= \varepsilon^{-1} v_σ(\varepsilon^{-1}x_σ)$, where $σ= 23, 12, 31$ is an index that runs over all the possible pairings of the three particles, $x_σ$ is the relative coordinate between two particles, and $\varepsilon$ is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials $v_σ$ with $α_σδ_σ$, where $δ_σ$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_σ=0$ and $α_σ= \int_{\mathbb{R}} v_σdx_σ$. To prove the convergence of the resolvents we make use of Faddeev's equations.

math-ph↗

arXiv2017

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation \begin{equation*} i\frac{\partial }{\partial t} ψ^\varepsilon(t)= -Δψ^\varepsilon(t) + g(\varepsilon,μ,|(ρ^\varepsilon,ψ^\varepsilon(t))|^{2μ}) (ρ^\varepsilon,ψ^\varepsilon(t)) ρ^\varepsilon \end{equation*} where $ρ^{\varepsilon} \to δ_0$ weakly and the function $g$ embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.

math-ph↗

arXiv2017

Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains

We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are semi-infinite straight strips. We make the waveguide collapse onto a graph by squeezing the edge regions to half-lines and the vertex region to a point. In a setting in which the ratio between the width of the waveguide and the longitudinal extension of the vertex region goes to zero, we prove the convergence of the operator to a selfadjoint realization of the Laplacian on a two edged graph. In the limit operator, the boundary conditions in the vertex depend on the spectral properties of an effective one dimensional Hamiltonian associated to the vertex region.

math-ph↗

arXiv2017

Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes

We study the relative zeta function for the couple of operators $A_0$ and $A_α$, where $A_0$ is the free unconstrained Laplacian in $L^2(\mathbf{R}^d)$ ($d \geq 2$) and $A_α$ is the singular perturbation of $A_0$ associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter $α$, which is related to the strength of the perturbation, is of the kind $α=α(-Δ_{\parallel})$, where $-Δ_{\parallel}$ is the free Laplacian in $L^2(\mathbf{R}^{d-1})$. Thus $α$ may depend on the components of the wave vector parallel to hyperplane; in this sense $A_α$ describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever $α=χ_{I}(-Δ_{\parallel})$, where $χ_{I}$ is the characteristic function of an interval $I$, the thermal Casimir energy can be explicitly computed.

math-ph↗

arXiv2016

Self-adjoint indefinite Laplacians

Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mathcal{L}_μ$ simply coincides with $L_μ$ and has compact resolvent. If $n=2$, then $\mathcal{L}_1$ has a non-empty essential spectrum, $σ_\mathrm{ess}(\mathcal{L}_{1})=\{0\}$. If $n\ge 3$, the spectral properties of $\mathcal{L}_1$ depend on the geometry of $Σ$. In particular, it has compact resolvent if $Σ$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Σ$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Σ$. We discuss some links between the resulting self-adjoint operator $\mathcal{L}_μ$ and some effects observed in negative-index materials.

math.SP↗

arXiv2016

Relative partition function of Coulomb plus delta interaction

The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.

math-ph↗