Normalized NLS ground states on a double plane hybrid

TL;DR

This paper studies the existence and properties of normalized ground states for a nonlinear Schrödinger equation on a double-plane hybrid, revealing how ground states behave depending on the matching conditions and parameters.

Contribution

It provides a comprehensive analysis of ground states on a double-plane quantum hybrid, including existence, qualitative behavior, and mass distribution depending on boundary conditions.

Findings

Ground states exist for all masses.

States can concentrate on one plane or distribute on both.

Logarithmic singularities appear at the origin.

Abstract

We investigate the existence and the properties of normalized ground states of a nonlinear Schr\"odinger 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.