Margherita Nolasco

Multiple homoclinic solutions for a class of autonomous singular systems in R2

Abstract We look for homoclinic solutions for a class of second order autonomous Hamiltonian systems in R 2 with a potential V having a strict global maximum at the origin and a finite set S ⊂ R 2 of singularities, namely V ( x ) → −∞ as dist( x , S ) → 0. We prove that if V satisfies a suitable geometrical property then for any k ∈ N the system admits a homoclinic orbit turning k times around a singularity ξ ∈ S .

Asymptotics for selfdual vortices on the torus and on the plane : A gluing technique

We consider multivortex solutions for the selfdual Abelian Higgs model, as the ratio of the vortex core size to the separation distance between vortex points (the scaling parameter) tends to zero. To this end, we use a gluing technique (a shadowing lemma) for solutions to the corresponding semilinear elliptic equation on the plane, allowing any number (finite or countable) of arbitrarily prescribed singular sources. Our approach is particularly convenient and natural for the study of the asymptotics. In particular, in the physically relevant cases where the vortex points are either finite or periodically arranged in the plane, we prove that a frequently used factorization ansatz for multivortex solutions is rigorously satisfied, up to an error which is exponentially small.

A Variational Approach to the Eigenfunctions of the One Particle Relativistic Hamiltonian

In this note we give a variational characterization of the eigenvalues and eigenvectors for the operator

Topological observables' in semiclassical field theories

H = {H_0} + V = \sqrt { - {c^2}\Delta + {m^2}{c^4}} + V,

On a Sharp Sobolev‐Type Inequality on Two-Dimensional Compact Manifolds

H=H0+V=−c2Δ+m2c4+V, where H0) is the relativistic (free) Hamiltonian operator and V is a real valued potential. Our results hold when

Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations

V(x) = \frac{1}{{\left| x \right|}}