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Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions

Vladimir Lotoreichik, Alessandro Michelangeli

Abstract

We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrödinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.

Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions

Abstract

We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrödinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.

Paper Structure

This paper contains 10 sections, 15 theorems, 86 equations.

Table of Contents

  1. Background and outline
  2. Preparatory materials
  3. Symmetric decreasing rearrangement
  4. Green functions
  5. Dirichlet Laplacian on bounded domain with point interaction
  6. The lowest eigenvalue and the ground state of ${\mathsf H}_{\alpha,y}^\Omega$
  7. The Faber-Krahn inequality for the operator ${\mathsf H}_{\alpha,y}^\Omega$
  8. Systems with Coulomb interactions
  9. One-body case
  10. Two-body case

Key Result

Lemma 2.1

Let $u\colon {\mathbb R}^d\rightarrow{\mathbb R}$, $d \ge 2$, be a non-negative measurable function vanishing at infinity. Let ${\mathcal{A}}\subset{\mathbb R}^d$ be a measurable set of finite volume. Then:

Theorems & Definitions (29)

  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • ...and 19 more