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Normalized NLS ground states on a double plane hybrid

Filippo Boni, Raffaele Carlone, Ilenia Di Giorgio

TL;DR

This work analyzes normalized ground states for a nonlinear Schrödinger equation on a double-plane quantum hybrid formed by two planes joined at a point. By incorporating plane-specific powers p_i, point-interaction parameters σ_i, and a quadratic inter-plane coupling through the charges q_i, the authors establish existence of ground states for any mass μ>0 and reveal two regimes depending on the coupling parameter β: decoupled (β=0) ground states localize on a single plane, while coupled (β>0) states occupy both planes with positive charges, positive, radially symmetric regular parts, and logarithmic singularities at the junction. A key technical achievement is the development of modified Gagliardo–Nirenberg inequalities on the point-interaction domain, enabling precise control of L^p norms in the variational analysis. The paper also analyzes how mass and logarithmic singularities distribute across the two planes as the matching parameters σ_i and the nonlinear powers p_i vary, including asymptotic regimes and μ^* thresholds. These results extend the understanding of nonlinear dispersive equations on non-standard domains and provide detailed insights into mass distribution in quantum hybrid geometries.

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.

Normalized NLS ground states on a double plane hybrid

TL;DR

This work analyzes normalized ground states for a nonlinear Schrödinger equation on a double-plane quantum hybrid formed by two planes joined at a point. By incorporating plane-specific powers p_i, point-interaction parameters σ_i, and a quadratic inter-plane coupling through the charges q_i, the authors establish existence of ground states for any mass μ>0 and reveal two regimes depending on the coupling parameter β: decoupled (β=0) ground states localize on a single plane, while coupled (β>0) states occupy both planes with positive charges, positive, radially symmetric regular parts, and logarithmic singularities at the junction. A key technical achievement is the development of modified Gagliardo–Nirenberg inequalities on the point-interaction domain, enabling precise control of L^p norms in the variational analysis. The paper also analyzes how mass and logarithmic singularities distribute across the two planes as the matching parameters σ_i and the nonlinear powers p_i vary, including asymptotic regimes and μ^* thresholds. These results extend the understanding of nonlinear dispersive equations on non-standard domains and provide detailed insights into mass distribution in quantum hybrid geometries.

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 -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.
Paper Structure (10 sections, 11 theorems, 131 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and main results.
  3. Preliminaries
  4. About the representation of the domain $\mathcal{D}$
  5. The problem at infinity on the hybrid.
  6. Modified Gagliardo-Nirenberg inequalities
  7. Proof of Theorem \ref{['thm:existence']}
  8. Properties of ground states: proof of Theorem \ref{['thm:charact']}
  9. Proof of Theorems \ref{['thm:p1=p2']} and \ref{['thm:sigma1=sigma2']}
  10. The self-adjoint operator

Key Result

Theorem 1.2

For every $\mu>0$, there exists at least a ground state of eq:F at mass $\mu$.

Figures (1)

  • Figure 1: Halfline-plane hybrid (on the left) and line-plane hybrid (on the right).

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2: Existence of ground states
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 15 more