Page 465 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 465
ANALYSIS OF PDES ON NETWORKS (MS-26)
nonlinearity, namely
Dψ − χK|ψ|p−2 ψ = ωψ , ψ : G → C2 , p > 2 , (2)
where G is a metric graph with finitely many edges and K = ∅ is its compact core, i.e. the set
of bounded edges, and χK is its characteristic function. The reduction to this simplified model
arises if one assumes that the nonlinearity affects only the compact core of the graph. This idea
was originally exploited in the case of Schrödinger equation in and it represents a preliminary
step toward the investigation of the case with the “extended” nonlinearity.
Considering the operator endowed with Kirchoff-type vertex conditions, we proved that (2)
possesses infinitely many solutions with ω ∈ (−mc2, mc2), converging, after a suitable renor-
malization, in H1-sense to solutions to the analogous Schrödinger equation −u − χK|u|p−2u =
λu for some λ < 0 ., in the non-relativistic limit as c → +∞.
Dynamics and scattering of truncated coherent states on the star-graph in
the semiclassical limit
Claudio Cacciapuoti, claudio.cacciapuoti@uninsubria.it
Università degli Studi dell’Insubria, Italy
Coauthors: Davide Fermi, Andrea Posilicano
We consider the dynamics of a quantum particle of mass m on the star-graph constituted by
n half-lines with a common origin. The generator of the dynamics is the Hamiltonian HK =
−(2m)−1 2∆ with Kirchhoff conditions in the vertex, is the reduced Planck constant. Our
aim is to obtain the semiclassical limit of the quantum evolution, generated by HK, of an initial
state resembling a coherent state (gaussian packet) concentrated on one of the edges of the
graph. Due to the Kirchhoff conditions in the vertex, the corresponding classical dynamics on
the graph cannot be described by Hamilton-Jacobi equations. For this reason, we define the
classical dynamics through a Liouville operator on the graph, obtained by means of the Krein’s
theory of singular perturbations of self-adjoint operators. For the same class of initial states, we
study also the semiclassical limit of the wave and scattering operators for the couple HK and
HD, where HD is the free Hamiltonian with Dirichlet conditions in the vertex.
On the nonlinear Dirac equation on noncompact metric graphs
Raffaele Carlone, raffaele.carlone@unina.it
Università Federico II Napoli, Italy
Coauthors: William Borrelli, Lorenzo Tentarelli
We discuss the Nonlinear Dirac Equation, with Kerr-type nonlinearity, on non-compact metric
graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. We will
present results about the well-posedness for the associated Cauchy problem in the operator
domain and, for infinite N-star graphs, and the existence of standing waves.
463
nonlinearity, namely
Dψ − χK|ψ|p−2 ψ = ωψ , ψ : G → C2 , p > 2 , (2)
where G is a metric graph with finitely many edges and K = ∅ is its compact core, i.e. the set
of bounded edges, and χK is its characteristic function. The reduction to this simplified model
arises if one assumes that the nonlinearity affects only the compact core of the graph. This idea
was originally exploited in the case of Schrödinger equation in and it represents a preliminary
step toward the investigation of the case with the “extended” nonlinearity.
Considering the operator endowed with Kirchoff-type vertex conditions, we proved that (2)
possesses infinitely many solutions with ω ∈ (−mc2, mc2), converging, after a suitable renor-
malization, in H1-sense to solutions to the analogous Schrödinger equation −u − χK|u|p−2u =
λu for some λ < 0 ., in the non-relativistic limit as c → +∞.
Dynamics and scattering of truncated coherent states on the star-graph in
the semiclassical limit
Claudio Cacciapuoti, claudio.cacciapuoti@uninsubria.it
Università degli Studi dell’Insubria, Italy
Coauthors: Davide Fermi, Andrea Posilicano
We consider the dynamics of a quantum particle of mass m on the star-graph constituted by
n half-lines with a common origin. The generator of the dynamics is the Hamiltonian HK =
−(2m)−1 2∆ with Kirchhoff conditions in the vertex, is the reduced Planck constant. Our
aim is to obtain the semiclassical limit of the quantum evolution, generated by HK, of an initial
state resembling a coherent state (gaussian packet) concentrated on one of the edges of the
graph. Due to the Kirchhoff conditions in the vertex, the corresponding classical dynamics on
the graph cannot be described by Hamilton-Jacobi equations. For this reason, we define the
classical dynamics through a Liouville operator on the graph, obtained by means of the Krein’s
theory of singular perturbations of self-adjoint operators. For the same class of initial states, we
study also the semiclassical limit of the wave and scattering operators for the couple HK and
HD, where HD is the free Hamiltonian with Dirichlet conditions in the vertex.
On the nonlinear Dirac equation on noncompact metric graphs
Raffaele Carlone, raffaele.carlone@unina.it
Università Federico II Napoli, Italy
Coauthors: William Borrelli, Lorenzo Tentarelli
We discuss the Nonlinear Dirac Equation, with Kerr-type nonlinearity, on non-compact metric
graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. We will
present results about the well-posedness for the associated Cauchy problem in the operator
domain and, for infinite N-star graphs, and the existence of standing waves.
463
