arXiv subjects

Diego Noja

Publications and source records attributed to Diego Noja.

At least 19 recordsLinked to original sources

arXiv2026

Robin Laplacians with point interactions on unbounded domains: discrete spectrum and coupling asymptotics

We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.

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arXiv2026

The Dirichlet Laplacian with a point interaction on unbounded Lipschitz domains

We study one-centre point interactions for the Dirichlet Laplacian on unbounded domains in dimensions two and three, with emphasis on exterior domains and special Lipschitz domains. These operators are singular perturbations constructed as self-adjoint extensions of the Dirichlet Laplacian restricted to functions vanishing at the interaction centre, and their resolvents are given by an explicit Kre\u{\i}n formula with a single extension parameter $\alpha$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $\alpha$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kre\u{\i}n formula. We establish domain monotonicity of the Weyl function, of the critical coupling, and of the unique negative eigenvalue when existing, and we derive sharp near-boundary asymptotics of the critical coupling in uniformly $C^{1,1}$ geometries. These estimates imply that, for every fixed coupling, nonpositive spectrum disappears when the interaction centre approaches the Dirichlet boundary. We also prove limiting absorption principles and purely absolutely continuous positive spectrum for a point-interaction in exterior domains case and in classes of special Lipschitz domains. We also analyze in depth several threshold phenomena. We show that the critical coupling is governed by the far-field behavior of the zero-energy Green function: exterior domains give threshold resonances, domains contained in a three-dimensional half-space give threshold eigenvalues, the half-plane gives a $p$-wave resonance, and planar wedges exhibit types of threshold states that are aperture-dependent. Finally, low-energy resolvent expansions are computed in the model cases and persistence or disappearance of eigenvalues at threshold are studied. The present paper seems to be the first systematic work on the subject.

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arXiv2026

Confinement and orbital stability of solitons of the NLS equation on metric graphs

We study the behavior of soliton states for the subcritical, time-dependent focusing NLS equation on a large family of non-compact metric graphs with Kirchhoff boundary conditions. This family is characterized by a topological assumption (``Assumption H'' in the literature) which rules out the existence of a ground state for all members of the class, with a single exception: the bubble-tower metric graph. We present two main results. First, we show that if the initial datum is close (in the energy norm) to a soliton placed on a single half-line of the graph and sufficiently far from the nearest vertex, then the corresponding solution remains confined to the same half-line for all times, and close to the soliton, up to a remainder that stays small in the energy norm. As a nontrivial application, this yields reflection of a slow soliton upon collision with the compact core of the graph, a phenomenon that first we prove and then we further investigate numerically. Second, for the exceptional case of bubble-tower graphs, we prove that the ground state (which exists only in this case) is orbitally stable. We emphasize that this example does not allow an immediate application of the Cazenave--Lions orbital stability argument, which requires a suitable modification. Finally, we discuss how the ideas and methods developed here may extend beyond the class of metric graphs with Kirchhoff boundary conditions and satisfying Assumption H. In particular, we extend the results to the meaningful case of the line in the presence of a smooth potential or a delta interaction.

math.AP↗

arXiv2026

Point interactions and singular solutions to semilinear elliptic equations

We investigate the connection between semilinear elliptic PDEs with isolated singularities and stationary nonlinear Schr\"odinger equations with point interactions. In dimensions $d=2,3$, we establish a rigorous correspondence between their solutions, revealing two regimes depending on whether a boundary condition at the singularity can be imposed. This connection enables us to exploit operator-theoretic and variational methods that have not previously been applied to the study of isolated singularities. In the source regime, we prove the existence of infinitely many radial singular solutions, by applying the symmetric mountain pass theorem of Ambrosetti and Rabinowitz to the action functional associated with the point interaction. When $d=2$, a suitable uniqueness result allows us to characterize singular ground states (positive solutions) as action minimizers and to prove the existence of infinitely many nodal singular solutions.

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arXiv2025

Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields

We characterise the families of self-adjoint Dirac and Schrödinger operators with Aharonov-Bohm magnetic field, and we exploit the non-relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short-scale boundary condition of self-adjointness. This ensures that the scattering length of the Aharonov-Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac-AB operators is mapped, in the non-relativistic limit, into the physically relevant sub-family of $s$-wave, angular-momentum-commuting, Schrö\-dinger-AB Hamiltonians with relativistic Dirac approximants.

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arXiv2024

Approximation of Schr\"odinger operators with point interactions on bounded domains

We consider Schr\"odinger operators on a bounded domain $\Omega\subset \mathbb{R}^3$, with homogeneous Robin or Dirichlet boundary conditions on $\partial\Omega$ and a point (zero-range) interaction placed at an interior point of $\Omega$. We show that, under suitable spectral assumptions, and by means of an extension-restriction procedure which exploit the already known result on the entire space, the singular interaction is approximated by rescaled sequences of regular potentials. The result is missing in the literature, and we also take the opportunity to point out some general issues in the approximation of point interactions and the role of zero energy resonances.

math-ph↗

arXiv2024

Resonances and resonance expansions for point interactions on the half-space

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space $\Omega =\mathbb R^3_+$ given by the self-adjoint operator named $\delta$-interaction. We will assume Dirichlet or Neumann boundary conditions on $\partial \Omega$. At variance with the well known case of $\mathbb R^3$, the resonances constitute an infinite set, here completely characterized. Moreover, we prove that resonances have an asymptotic distribution satisfying a modified Weyl law and we give the semiclassical asymptotics. Finally we give applications of the results to the asymptotic behavior of the abstract wave and Schr\"odinger dynamics generated by the Laplacian with a point interaction on the half-space

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arXiv2024

On solitary waves for the Korteweg--de Vries equation on metric star graphs

We study the Korteweg--de Vries equation on a metric star graph and investigate existence of solitary waves on the metric graph in terms of the coefficients of the equation on each edge, the coupling condition at the central vertex of the star and the speeds of the travelling wave. We show that, with a continuity condition at the vertex, solitary waves can occur exactly when the parameters are chosen in a fairly special manner. We also consider coupling conditions beyond continuity.

math.AP↗

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

Standing waves on quantum graphs

We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.

math.AP↗

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

Standing waves of the quintic NLS equation on the tadpole graph

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $ω\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $ω\in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $ω_0$ and $ω_1$ such that the standing waves are the ground states for $ω\in [ω_0,0)$, local minimizers of the energy at constant mass for $ω\in (ω_1,ω_0)$, and saddle points of the energy at constant mass for $ω\in (-\infty,ω_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.

math.AP↗

arXiv2019

A Dirac field interacting with point nuclear dynamics

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in $H^s$ with $s>1$ and local well posedness in $H^s$ for $s>3/2$. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

math.AP↗

arXiv2017

Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy

We study a boundary value problem related to the search of standing waves for the nonlinear Schrödinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the {\it double-bridge graph}, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-called Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$, and we are interested in the existence and properties of standing waves of given frequency $ω$. For every $ω>0$ only solutions supported on the circle exist (cnoidal solutions), and only for a rational value of $L_1/L_2$; they can be extended to every $ω\in \mathbb{R}$. We study, for $ω<0$, the solutions periodic on the circle but with nontrivial components on the half-lines. The problem turns out to be equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter $ω$. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of frequencies $ω_n$. Moreover we show that the frequency sequence $\{ω_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and it admits at least a finite limit point, in general non-zero. Finally, any negative real number can be a limit point of a set of admitted frequencies up to the choice of a suitable irrational geometry $L_1/L_2$ for the graph. These results depend on basic properties of diophantine approximation of real numbers.

math.AP↗

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↗

arXiv2016

Ground state and orbital stability for the NLS equation on a general starlike graph with potentials

We consider a nonlinear Schrödinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of $δ$-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass ($L^2$-norm). When existing, the minimizer is called ground state and it is the profile of an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists for sufficiently small masses whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state). This is a wide generalization of a result previously obtained for a star graph with a single vertex. The main part of the proof is devoted to prove the concentration compactness principle for starlike structures; this is non trivial due to the lack of translation invariance of the domain. Then we show that a minimizing bounded $H^1$ sequence for the constrained NLS energy with external linear potentials is in fact convergent if its mass is small enough. Examples are provided with discussion of hypotheses on the linear part.

math-ph↗

arXiv2016

Airy-type evolution equations on star graphs

In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its correct definition, with different characterizations, as a skew-adjoint operator on a star graph, a set of lines connecting at a common vertex representing, for example, a network of branching channels. A necessary condition turns out to be that the graph is balanced, i.e. there is the same number of ingoing and outgoing edges at the vertex. The simplest example is that of the line with a point interaction at the vertex. In these cases the Airy dynamics is given by a unitary or isometric (in the real case) group. In particular the analysis provides the complete classification of boundary conditions giving momentum (i.e., $L^2$-norm of the solution) preserving evolution on the graph. A second more general problem here solved is the characterization of conditions under which the Airy operator generates a contraction semigroup. In this case unbalanced star graphs are allowed. In both unitary and contraction dynamics, restrictions on admissible boundary conditions occur if conservation of mass (i.e., integral of the solution) is further imposed. The above well posedness results can be considered preliminary to the analysis of nonlinear wave propagation on branching structures.

math-ph↗

arXiv2016

The 1-D Dirac equation with concentrated nonlinearity

We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-Šeba models and the concentrated versions of the Bragg Resonance, Gross-Neveu, and Soler type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable (the "charge").

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