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On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

Thiago Carvalho Corso, Tobias Ried

TL;DR

This work resolves the existence and explicit form of optimizers for a variational problem that governs upper bounds in the CLR and LT inequalities, recasting the problem via duality into a three-lines framework on Hardy-type spaces over a strip. The authors derive a complete dual formulation, then construct the optimizer analytically as a Blaschke factor times an exponential phase, yielding a closed-form expression for the optimal constant $M_\gamma$ and enabling high-precision computation. The results yield improved upper bounds for CLR/Lie-Thirring constants in many regimes (notably large $d/\sigma$) and provide sharp asymptotics as $\gamma\downarrow 2$ and $\gamma\to\infty$, with explicit formulas and numerical tables. The methodology blends variational analysis in $L^1$, complex analysis on strips, and Euler–Lagrange equations, highlighting a deep connection between Fourier transform norms, Hardy spaces, and spectral bounds for Schrödinger-type operators.

Abstract

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We achieve this through a variational characterization of the $L^1$ norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].

On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

TL;DR

This work resolves the existence and explicit form of optimizers for a variational problem that governs upper bounds in the CLR and LT inequalities, recasting the problem via duality into a three-lines framework on Hardy-type spaces over a strip. The authors derive a complete dual formulation, then construct the optimizer analytically as a Blaschke factor times an exponential phase, yielding a closed-form expression for the optimal constant and enabling high-precision computation. The results yield improved upper bounds for CLR/Lie-Thirring constants in many regimes (notably large ) and provide sharp asymptotics as and , with explicit formulas and numerical tables. The methodology blends variational analysis in , complex analysis on strips, and Euler–Lagrange equations, highlighting a deep connection between Fourier transform norms, Hardy spaces, and spectral bounds for Schrödinger-type operators.

Abstract

We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We achieve this through a variational characterization of the norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].
Paper Structure (19 sections, 29 theorems, 201 equations, 2 tables)

This paper contains 19 sections, 29 theorems, 201 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications to CLR and LT inequalities
  4. Outline of the proofs of our main theorems
  5. Overview of the article
  6. The Dual problem
  7. Formulation over L1(R)
  8. Duality
  9. Reformulation in the complex plane
  10. Mixed Hardy-like spaces on the strip
  11. The variational problem over holomorphic functions
  12. Characterization of optimizers
  13. Euler-Lagrange equation and dual-primal optimizer relation
  14. Optimizers
  15. Further consequences for the CLR and LT bound
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $\gamma >2$ and $M_\gamma$ be defined via eq:primal. Then we have where $h_0$ and $h_{-1}$ are the boundary values of $h$ in the sense described above. Moreover, the maximizer of eq:3linesproblem exists and is unique up to the transformation $h_{\alpha,\beta,\omega}(z) = \beta h(z-\alpha)\mathrm{e}^{\mathrm{i}\omega z}$ for $\alpha,\omega \in \mathbb R$ and $\beta

Theorems & Definitions (63)

  • Definition 1.1: $\mathbb{H}^{p,q}(S)$ spaces
  • Theorem 1.2: Three lines variational problem
  • Theorem 1.3: Optimizer
  • Theorem 1.4: CLR bound
  • Remark 1.5: Asymptotics of CLR upper bound
  • Theorem 1.6: LT Bound
  • Corollary 1.7: LT Bound Updated
  • Lemma 2.1: Fourier transform of integrable functions
  • Remark
  • Remark : Connection with maximal Fourier multipliers
  • ...and 53 more