arXiv subjects

Élio Durand-Simonnet

Publications and source records attributed to Élio Durand-Simonnet.

4 recordsLinked to original sources

arXiv2026

Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials

We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.

math.AP↗

arXiv2026

On the defocusing stationary nonlinear Schrödinger equation on metric graphs

We study the defocusing nonlinear Schrödinger equation on noncompact metric graphs under general self-adjoint vertex conditions ensuring the existence of a negative eigenvalue of the Hamiltonian operator. First, we focus on the existence of energy ground states with prescribed mass. We show that existence and stability always hold for small masses and fail for large masses in the $L^2$-subcritical regime. For $δ$-type vertex conditions, we provide more precise results: ground states exist for all masses in the $L^2$-critical and supercritical cases, while in the subcritical case, for one vertex graphs, there exists a sharp mass threshold such that ground states exist below it and do not exist above it. Moreover, we show that the ground state bifurcates from the vanishing solution at the bottom of the Hamiltonian spectrum. Finally, we present multiplicity results for stationary solutions, both in the fixed-frequency and fixed-mass settings.

math.AP↗

arXiv2025

Ground States for the Defocusing Nonlinear Schrödinger Equation on Non-Compact Metric Graphs

We investigate the existence and stability of ground states for the defocusing nonlinear Schrödinger equation on non-compact metric graphs. We establish a sharp criterion for the existence of action ground states in terms of the spectral properties of the underlying Hamiltonian: ground states exist if and only if the bottom of the spectrum is negative and the frequency lies within a suitable range. We further explore the relation between action and energy ground states, showing that while every action minimizer yields an energy minimizer, the converse fails in general. In particular, we prove that energy ground states may not exist for arbitrary masses. This discrepancy is illustrated through explicit examples on star graphs with $δ$ and $δ'$-type vertex conditions: in the mass-subcritical case, we exhibit a large interval of masses for which no energy minimizer exists, whereas in the supercritical regime, energy ground states exist for all masses.

math.AP↗

arXiv2025

Ground States of the Nonlinear Schr{ö}dinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition

We consider the stationary nonlinear Schr{ö}dinger equation set on a tadpole graph with a repulsive delta vertex condition between the loop and the tail of the tadpole. We establish the existence of an action ground state when the size of the loop is either very small or very large. Our analysis relies on variational arguments, such as profile decomposition. When it exists, we study the shape of the ground state using ordinary differential equations arguments, such as the study of period functions. The theoretical results are completed with a numerical study.

math.AP↗