The nonlinear Schrödinger equation for orthonormal functions II. Applications to Lieb-Thirring inequalities
R.L. Frank, D. Gontier, M. Lewin
Commun. Math. Phys. 384, 1783- 1828 (2021).
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) are raised to the power κ is never given by the one-bound state case when κ>max(0,2−d/2) in space dimension d≥1. When in addition κ≥1 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.R. L. Frank, E. H. Lieb
The periodic Lieb-Thirring inequality
R.L. Frank, D. Gontier, M. Lewin
Book: Partial Differential Equations, Spectral theory and Mathematical Physics 135-154 (2021).
We discuss the Lieb–Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb–Thirring inequality at the special exponent γ=32 admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic.
A non-linear adiabatic theorem for the one-dimensional Landau-Pekar equations
R.L. Frank, Z. Gang
Journal of Functional Analysis 279 (7), 108631 (2020).
We discuss a one-dimensional version of the Landau-Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear adiabatic theorem. Dispersive estimates for solutions of the Schrodinger equation with time-dependent potential are a key technical ingredient in our proof. (C) 2020 Elsevier Inc. All rights reserved.
How much delocalisation is needed for an enhanced area law of the entanglement entropy?
Peter Müller, Leonid Pastur, Ruth Schulte
Commun. Math. Phys. 376 (1), 649 - 679 (2020).
We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficiently small disorder, we show that the entanglement entropy exhibits at least a logarithmically enhanced area law if the Fermi energy coincides with a critical energy of the model where the localisation length diverges.
Proof of the strong Scott conjecture for Chandrasekhar atoms
R.L. Frank, K. Merz, H. Siedentop, B. Simon
Pure and Applied Functional Analysis 5 (6), 1319 - 1356 (2020).
We consider a large neutral atom of atomic number Z, taking relativistic effects into account by assuming the dispersion relation √(c^2p^2+c^4). We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c→∞ keeping Z/c fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density.
Equivalence of Sobolev Norms Involving Generalized Hardy Operators
R.L. Frank, K. Merz, H. Siedentop
International Mathematics Research Notices 2021, 2284-2303 (2021).
We consider the fractional Schrödinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrödinger operators and have an important application in the treatment of large relativistic atoms.