##### Logarithmic estimates for mean-field models in dimension two and the Schrodinger-Poisson system

J. Dolbeault, R.L. Frank, L. Jeanjean

Comptes Rendus Mathematique 359, 1279-1293 (2021).

In dimension two, we investigate a free energy and the ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrodinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

##### Proof of spherical flocking based on quantitative rearrangement inequalities

R.L. Frank, E.H. Lieb

Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze 22 (3), 1241-1263 (2021).

Our recent work on the Burchard-Choksi-Topaloglu flocking problem showed that in the large mass regime the ground state density profile is the characteristic function of some set. Here we show that this set is, in fact, a round ball. The essential mathematical structure needed in our proof is a strict rearrangement inequality with a quantitative error estimate, which we deduce from recent deep results of M. Christ.

##### Existence and nonexistence in the liquid drop model

R.L. Frank, P.T. Nam

Calculus of Variations and Partial Differential Equations 60, 223 (2021).

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

##### 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

##### On the spectrum of the Kronig-Penney model in a constant electric field

R.L. Frank, S. Larson

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator

−d2dx2−Fx+∑n∈Zgnδ(x−n)in L2(R)

with F>0 and two different choices of the coupling constants {gn}n∈Z. In the first model gn≡λ and we prove that if F∈π2Q then the spectrum is R and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model gn are independent random variables with mean zero and variance λ2. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is R and it is dense pure point if F<λ2/2 and purely singular continuous if F>λ2/2.

##### 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.

##### 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 (2019).

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.

##### The BCS critical temperature in a weak magnetic field

R. Frank, C. Hainzl, E. Langmann

Journal of Spectral Theory 9 (3), 1005–1062 (2019).

We show that, within a linear approximation of BCS theory, a weak homogeneous magnetic field lowers the critical temperature by an explicit constant times the field strength, up to higher order terms. This provides a rigorous derivation and generalization of results obtained in the physics literature fromWHH theory of the upper critical magnetic field. A new ingredient in our proof is a rigorous phase approximation to control the effects of the magnetic field.

##### The BCS critical temperature in a weak external field via a linear two-body operator

R. Frank, C. Hainzl

Workshop on Macroscopic Limits of Quantum Systems 29-62 (2018).

We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in Frank et al. (Commun Math Phys 342(1):189–216, 2016) using the strategy introduced in Frank et al. (The BCS critical temperature in a weak homogeneous magnetic field), where we considered the case of an external constant magnetic field.