Page 467 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 467
ANALYSIS OF PDES ON NETWORKS (MS-26)
The quintic NLS on the tadpole graph
Diego Noja, diego.noja@unimib.it
Università di Milano Bicocca, Italy
Coauthor: Dmitry Pelinovsky
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 and Kirchhoff
boundary conditions at the vertex. The profile of a standing wave with frequency ω ∈ (−∞, 0)
is characterized as a global minimizer of the quadratic part of energy constrained to the unit
sphere in L6. The set of standing waves so defined strictly includes the set of ground states,
i.e. the global minimizers of the energy at constant mass (L2-norm), but it is actually wider.
While ground states exist only for a certain interval of masses, the above standing waves exist
for every ω ∈ (−∞, 0) and correspond to a bigger interval of masses. It is proven that there
exist critical frequencies ω1 and ω0 with −∞ < ω1 < ω0 < 0 such that the standing waves are
the ground state for ω ∈ [ω0, 0), local constrained minima of the energy for ω ∈ (ω1, ω0) and
saddle points of the energy at constant mass for ω ∈ (−∞, ω1).
Joint work with D.E. Pelinovsky.
On Pleijel’s nodal domain theorem for quantum graphs
Marvin Plümer, marvin.pluemer@fernuni-hagen.de
FernUniversität in Hagen, Germany
In this talk we present recent results on metric graph counterparts of Pleijel’s theorem on the
asymptotics of the number of nodal domains νn of the n-th eigenfunction(s) of a broad class of
operators on compact metric graphs, including Schrödinger operators with L1-potentials as well
as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of
the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. We
characterise the accumulation points of the sequence ( νn )n∈N, which are shown to form a finite
n
subset of (0, 1]. This extends the previously known result that νn ∼ n generically, for certain
realisations of the Laplacian, in several directions. In particular, we will see that the existence of
accumulation smaller than 1 is strictly related to the failure of the unique continuation principle
on metric graphs.
Finally we show that for (most) metric graphs – metric trees and general metric graphs
with at least one Dirichlet vertex – there exists an infinite sequence of generic eigenfunctions –
namely, eigenfunctions of multiplicity 1 that do not vanish in the graph’s vertices – of the free
Laplacian and infer that, in this case, 1 is always an accumulation point of νn . In order to do so
n
we introduce a new type of secular function.
The talk is based on joint work with Matthias Hofmann (Lisbon), James Kennedy (Lisbon),
Delio Mugnolo (Hagen) and Matthias Täufer (Hagen).
465
The quintic NLS on the tadpole graph
Diego Noja, diego.noja@unimib.it
Università di Milano Bicocca, Italy
Coauthor: Dmitry Pelinovsky
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 and Kirchhoff
boundary conditions at the vertex. The profile of a standing wave with frequency ω ∈ (−∞, 0)
is characterized as a global minimizer of the quadratic part of energy constrained to the unit
sphere in L6. The set of standing waves so defined strictly includes the set of ground states,
i.e. the global minimizers of the energy at constant mass (L2-norm), but it is actually wider.
While ground states exist only for a certain interval of masses, the above standing waves exist
for every ω ∈ (−∞, 0) and correspond to a bigger interval of masses. It is proven that there
exist critical frequencies ω1 and ω0 with −∞ < ω1 < ω0 < 0 such that the standing waves are
the ground state for ω ∈ [ω0, 0), local constrained minima of the energy for ω ∈ (ω1, ω0) and
saddle points of the energy at constant mass for ω ∈ (−∞, ω1).
Joint work with D.E. Pelinovsky.
On Pleijel’s nodal domain theorem for quantum graphs
Marvin Plümer, marvin.pluemer@fernuni-hagen.de
FernUniversität in Hagen, Germany
In this talk we present recent results on metric graph counterparts of Pleijel’s theorem on the
asymptotics of the number of nodal domains νn of the n-th eigenfunction(s) of a broad class of
operators on compact metric graphs, including Schrödinger operators with L1-potentials as well
as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of
the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. We
characterise the accumulation points of the sequence ( νn )n∈N, which are shown to form a finite
n
subset of (0, 1]. This extends the previously known result that νn ∼ n generically, for certain
realisations of the Laplacian, in several directions. In particular, we will see that the existence of
accumulation smaller than 1 is strictly related to the failure of the unique continuation principle
on metric graphs.
Finally we show that for (most) metric graphs – metric trees and general metric graphs
with at least one Dirichlet vertex – there exists an infinite sequence of generic eigenfunctions –
namely, eigenfunctions of multiplicity 1 that do not vanish in the graph’s vertices – of the free
Laplacian and infer that, in this case, 1 is always an accumulation point of νn . In order to do so
n
we introduce a new type of secular function.
The talk is based on joint work with Matthias Hofmann (Lisbon), James Kennedy (Lisbon),
Delio Mugnolo (Hagen) and Matthias Täufer (Hagen).
465
