Normalized NLS ground states on a double plane hybrid

Abstract

We investigate the existence and the properties of normalized ground states of a nonlinear Schrödinger equation on a quantum hybrid formed by two planes connected at a point. The nonlinearities are of power type and $L^2$-subcritical, while the matching condition between the two planes generates two point interactions of different strengths on each plane, together with a coupling condition between the two planes. We prove that ground states exist for every value of the mass and two different qualitative situations are possible depending on the matching condition: either ground states concentrate on one of the plane only, or ground states distribute on both the planes and are positive, radially symmetric, decreasing and present a logarithmic singularity at the origin of each plane. Moreover, we discuss how the mass distributes on the two planes and compare the strengths of the logarithmic singularities on the two planes when the parameters of the matching condition and the powers of the nonlinear terms vary.

21 pages, 2 figures

References in corpus (11)

• On global minimizers for a mass constrained problem
• Limits of Quantum Graph Operators With Shrinking Edges
• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• Airy-type evolution equations on star graphs
• Edge-localized states on quantum graphs in the limit of large mass
• Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• The search for NLS ground states on a hybrid domain: Motivations, methods, and results