arXiv subjects

Lorenzo Zanelli

Publications and source records attributed to Lorenzo Zanelli.

6 recordsLinked to original sources

arXiv2026

From Klein-Gordon-Wave to Schrödinger-Wave: a Normal Form Approach

We consider a Klein-Gordon-Wave system, describing the evolution of a massive field and a massless one interacting through a Yukawa-like coupling, and we explicitly derive its Hamiltonian normal form to first and second order. To the first-order approximation, the normal form results in a Schrödinger-Wave system, which reduces to the Schrödinger-Poisson one in the singular limit of vanishing perturbative parameter. The second-order approximation provides the successive corrections to the Schrödinger-Wave system, and is presented in order to show that higher-order approximations to all orders can be obtained by iterating our constructive procedure. The normal form technique adopted here formally extends the standard Birkhoff normal form procedure for harmonic oscillators to include a set of free particles in the unperturbed problem. The mathematical result obtained here might explain, for example, the "cooling" process of ultra-light dark matter, the approximate validity of the Schrödinger-Poisson system describing its dynamics and the long term conservation of the total dark matter mass.

math-ph↗

arXiv2025

Scaling of highly excited Schrödinger-Poisson eigenstates and universality of their rotation curves

This work provides a comprehensive numerical characterization of the excited spherically symmetric stationary states of the Schrödinger-Poisson problem. Through numerical computation of highly excited eigenstates, novel heuristic laws are proposed, which describe how their fundamental features scale with the excitation index $n$. Key characteristics of the eigenfunctions include: the effective support, which exhibits a parabolic dependence on the excitation index; the distances between adjacent nodes, whose pattern varies regularly with $n$; and the oscillation amplitude, which follows a power law with an exponent approaching $-1$ for large $n$. Based on the eigenfunctions, eigenvelocities are conveniently defined. They exhibit a mid-range oscillatory region with an average linear trend, whose slope approaches zero in the large $n$ limit; and they are characterized by heuristic scaling relationships with the excitation index $n$, revealing an intrinsic universal behavior.

math-ph↗

arXiv2021

Periodic coherent states decomposition and quantum dynamics on the flat torus

We provide a result on the coherent states decomposition for functions in $L^2 (\Bbb T^n)$ where $\Bbb T^n := (\Bbb R / 2π\Bbb Z)^n$. Suddenly, we study such a decomposition with respect to the quantum dynamics related to semiclassical elliptic Pseudodifferential operators, and we prove a related invariance result.

math-ph↗

arXiv2018

An homogenization approach for the inverse spectral problem of periodic Schrödinger operators

We study the inverse spectral problem for periodic Schrödinger opera\-tors of kind $- \frac{1}{2} \hbar^2 Δ_x + V(x)$ on the flat torus $\Bbb T^n := (\Bbb R / 2 π\Bbb Z)^n$ with potentials $V \in C^{\infty} (\Bbb T^n)$. We show that if two operators are isospectral for any $0 < \hbar \le 1$ then they have the same effective Hamiltonian given by the periodic homogenization of Hamilton-Jacobi equation. This result provides a necessary condition for the isospectrality of these Schrödinger operators. We also provide a link between our result and the spectral limit of quantum integrable systems.

math-ph↗

arXiv2013

On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation

In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2π\Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b \in S^m (\mathbb{T}^n \times \mathbb{R}^n)$. Subsequently, we consider $\mathrm{Op}^w_\hbar(H)$ when $H=\frac{1}{2} |η|^2 + V(x)$ where $V \in C^\infty (\Bbb T^n;\Bbb R)$ and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on $\Bbb T^n \times \Bbb R^n$ written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space $H^{1} (\mathbb{T}^n; \Bbb C)$ with phase functions in the class of Lipschitz continuous weak KAM solutions (of positive and negative type) of the Hamilton-Jacobi equation $\frac{1}{2} |P+ \nabla_x v_\pm (P,x)|^2 + V(x) = \bar{H}(P)$ for $P \in \ell \Bbb Z^n$ with $\ell >0$, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of $P+ \nabla_x v_\pm$.

math.AP↗

arXiv2010

Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator

We construct a family of Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton-Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

math-ph↗