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Two dimensional NLS ground states with attractive Coulomb potential and point interaction

Filippo Boni, Matteo Gallone

TL;DR

This work analyzes a two-dimensional focusing NLS with an attractive Coulomb potential and a point interaction, formulating the linear part via self-adjoint extensions and studying nonlinear ground states at fixed mass. Using a Green's function framework, Krein–Višik–Birman extension theory, and refined variational tools (modified Gagliardo–Nirenberg and Hardy-type inequalities, rearrangement arguments), the authors prove existence and detailed qualitative properties of ground states for $\nu<0$ and $2<p<4$, including a logarithmic singularity at the interaction site and radial symmetry. They also establish existence and structure of action minimizers on the Nehari manifold for $\omega>\omega_{\nu}$, enabling extension to the $L^{2}$-critical and supercritical regime via a variational reduction. The results illuminate the interplay of point interactions and Coulomb attraction in shaping bound states, and lay groundwork for orbital stability analysis and potential generalizations to repulsive Coulomb or higher dimensions.

Abstract

We study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schrödinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any L$^2$-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the L$^2$-critical and supercritical cases.

Two dimensional NLS ground states with attractive Coulomb potential and point interaction

TL;DR

This work analyzes a two-dimensional focusing NLS with an attractive Coulomb potential and a point interaction, formulating the linear part via self-adjoint extensions and studying nonlinear ground states at fixed mass. Using a Green's function framework, Krein–Višik–Birman extension theory, and refined variational tools (modified Gagliardo–Nirenberg and Hardy-type inequalities, rearrangement arguments), the authors prove existence and detailed qualitative properties of ground states for and , including a logarithmic singularity at the interaction site and radial symmetry. They also establish existence and structure of action minimizers on the Nehari manifold for , enabling extension to the -critical and supercritical regime via a variational reduction. The results illuminate the interplay of point interactions and Coulomb attraction in shaping bound states, and lay groundwork for orbital stability analysis and potential generalizations to repulsive Coulomb or higher dimensions.

Abstract

We study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schrödinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any L-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the L-critical and supercritical cases.
Paper Structure (16 sections, 26 theorems, 162 equations)

This paper contains 16 sections, 26 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Preliminaries
  4. Properties of $\theta_{\lambda,\nu}$ and the Green's function $\mathcal{G}_{\lambda,\nu}$
  5. Gagliardo-Nirenberg type inequalities
  6. Hardy-type inequalities
  7. Some properties of the symmetric decreasing rearrangement
  8. Minimizers of the action: proof of Theorem \ref{['ex-min-act']}
  9. Fixed mass ground states: proof of Theorem \ref{['ex-gs-mu']}
  10. Self-adjoint extensions of $2d$ Schrödinger operator with Coulomb potential
  11. Unitary equivalent problem.
  12. Kernel of the shifted operator
  13. Domain of the adjoint
  14. Self-adjoint extensions of the radial operator.
  15. Reconstruction of the 2D problem
  16. ...and 1 more sections

Key Result

Proposition 1.1

For any $\nu\in \mathbb{R}$, the symmetric operator $H_{\nu}$ defined in eq:sym_vero admits a one-parameter family of self-adjoint extensions $\{H_{\nu,\alpha}\}_{\alpha \in \mathbb{R} \cup \{\infty\}}$. For any $\alpha\in \mathbb{R}$:

Theorems & Definitions (57)

  • Proposition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 47 more