2020/02/29 by Rupert L. Frank, David Gontier, Mathieu Lewin · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Conjecture #Constant (computer programming) #Dimension (graph theory) #Eigenvalues and eigenvectors #Nonlinear Partial Differential Equations #Nonlinear system #Operator (biology) #Orthonormal basis #Space (punctuation) #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #math.SP
paper · pdf · doi:10.1007/s00220-021-04039-5
Includes some new properties of the one-bound state (Gagliardo-Nirenberg) constant
openalex created_date 2020/02/24 · arxiv created 2020/09/04 · openalex publication_date 2021/05/18 · arxiv updated 2021/06/02 · openalex updated_date 2026/08/05
Abstract In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator -Δ +V(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are raised to the power κ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> is never given by the one-bound state case when κ gt;max (0,2-d/2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>></mml:mo> <mml:mo>max</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in space dimension d≥ 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . When in addition κ ≥ 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7 ). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.