Work

On a Variational Problem Related to the Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities

Year: 2025

Type: article

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 Hundertmark et al. (Invent Math 231:111–167, 2023. https://doi.org/10.1007/s00222-022-01144-7 ) and Frank et al. (Eur Math Soc 23(8):2583–2600, 2021. https://doi.org/10.1090/pspum/104/01877 ). We achieve this through a variational characterization of the $$L^1$$ 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 Hundertmark et al. (Invent Math 231:111–167, 2023. https://doi.org/10.1007/s00222-022-01144-7 ) and Frank et al. (Eur Math Soc 23(8):2583–2600, 2021. https://doi.org/10.1090/pspum/104/01877 ).

Source: Communications in Mathematical Physics

DOI: 10.1007/s00220-024-05216-y

Authors: Thiago Carvalho Corso, Tobias Ried

Institutions: University of Stuttgart, Max Planck Institute for Mathematics in the Sciences

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

Funder: Deutsche Forschungsgemeinschaft