##### DISCRETE SCHRODINGER OPERATORS WITH DECAYING AND OSCILLATING POTENTIALS

R. L. Frank, S. Larson

St Petersburg Mathematical Journal 35 (1), 233-244 (2024).

The paper is devoted to a family of discrete one-dimensional Schodingeroperators with power-like decaying potentials with rapid oscillations. In particular, the potential V(n)=lambda n(-alpha)cos(pi omega n beta)with1<beta<2 alpha, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian

##### Hardy inequalities for large fermionic systems

R. L. Frank, T. Hoffmann-Ostenhof, A. Laptev, J. P. Solovej

Journal of Spectral Theory 14 (2), 805-835 (2024).

Given 0<s<(d)/(2 )with s <= 1, we are interested in the large N-behavior of the optimal constant kappa N in the Hardy inequality ∑(N)(n=1)(-Delta(n))(s)>=kappa(N)∑(n<m)|X-n-X-m|(-2s), when restricted to antisymmetric functions. We show that N1-2s/d kappa(N) has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.

##### Endpoint Schatten class properties of commutators

R. L. Frank, F. Sukochev, D. Zanin

Advances in Mathematics 450, 109738 (2024).

We study the trace ideal properties of the commutators [( - Delta) is an element of/2 , M- f ] of a power of the Laplacian with the multiplication operator by a function f on R- d . For a certain range of is an element of is an element of R, we show that this commutator belongs to the weak L Schatten class L d/ 1 - is an element of , infinity if and only if the distributional gradient of f belongs to L d/1-is an element of . Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).

##### WEIGHTED CLR TYPE BOUNDS IN TWO DIMENSIONS

R. L. Frank, A. Laptev, L. Read

Transactions of the American Mathematical Society 15 (2024).

. We derive weighted versions of the Cwikel-Lieb-Rozenblum inequality for the Schro center dot dinger operator in two dimensions with a nontrivial Aharonov-Bohm magnetic field. Our bounds capture the optimal dependence on the flux and we identify a class of long-range potentials that saturate our bounds in the strong coupling limit. We also extend our analysis to the twodimensional Schro center dot dinger operator acting on antisymmetric functions and obtain similar results.

##### On Sobolev norms involving Hardy operators in a half-space

R. L. Frank, K. Merz

Journal of Functional Analysis 285 (10), 110104 (2023).

We consider Hardy operators on the half-space, that is, ordinary and fractional Schrodinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces generated by the Hardy operators and by the fractional Laplacian are comparable with each other when the coupling constant is not too large in a quantitative sense. Our results extend those in the whole Euclidean space and rely on recent heat kernel bounds.

##### Sharp inequalities for coherent states and their optimizers

R. L. Frank

Advanced Nonlinear Studies 23 (1), 20220050 (2023).

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Holder inequalities for polynomials, conjectured by Bodmann.

##### The Scott conjecture for large Coulomb systems: a review

R. L. Frank, K. Merz, H. Siedentop

Letters in Mathematical Physics 113 (1), 11 (2023).

We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.

##### ASYMPTOTICS OF SINGULAR VALUES FOR QUANTUM DERIVATIVES

R. L. Frank, F. Sukochev, D. Zanin

Transactions of the American Mathematical Society 42 (2023).

We obtain Weyl type asymptotics for the quantised derivative d over bar f of a function f from the homgeneous Sobolev space Wd1 (Rd) on Rd. The asymptotic coefficient II backward difference fIILd(Rd) is equivalent to the norm of d over bar f in the principal ideal Ld,infinity, thus, providing a non-asymptotic, uniform bound on the spectrum of d over bar f. Our methods are based on the C*-algebraic notion of the principal symbol mapping on Rd, as developed recently by the last two authors and collaborators.

##### SCHRODINGER OPERATORS: EIGENVALUES AND LIEB-THIRRING INEQUALITIES

R. L. Frank, A. Laptev, T. Weidl

SCHRODINGER OPERATORS: Eigenvalues and Lieb-Thirring Inequalities 200, Cambridge University Press, (2022).

We provide a brief, but self-contained, introduction to the theory of self-adjoint operators. In a first section we give the relevant definitions, including that of the spectrum of a self-adjoint operator, and we discuss the proof of the spectral theorem. In a second section, we discuss the connection between lower semibounded, self-adjoint operators and lower semibounded, closed quadratic forms, and we derive the variational characterization of eigenvalues in the form of Glazman’s lemma and of the Courant–Fischer–Weyl min-max principle. Furthermore, we discuss continuity properties of Riesz means and present in abstract form the Birman–Schwinger principle.

##### Degenerate stability of some Sobolev inequalities

R. L. Frank

Annales De L Institut Henri Poincare-Analyse Non Lineaire 39 (6), 1459-1484 (2022).

./ Abstract. We show that on S1(1= d -2) x Sd-1(1) the conformally invariant Sobolev inequal-ity holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is moti-vated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on Sd. Our proof proceeds by an iterated Bianchi-Egnell strategy.

##### An inequality for the normal derivative of the Lane-Emden ground state

R. L. Frank, S. Larson

Advances in Calculus of Variations 22 (2022).

We consider Lane-Emden ground states with polytropic index 0 <= q - 1 <= 1, that is, minimizers of the Dirichlet integral among L-q-normalized functions. Our main result is a sharp lower bound on the L-2-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Omega subset of R-d, without assuming convexity.

##### Which magnetic fields support a zero mode?

R. L. Frank, M. Loss

Journal Fur Die Reine Und Angewandte Mathematik 2022 (788), 1-36 (2022).

This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrodinger equation in three dimensions to have a negative energy bound state, the 3/2 norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3/2 norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.

##### Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions

J. A. Carrillo, M. G. Delgadino, R. L. Frank, M. Lewin

Mathematical Models & Methods in Applied Sciences 32 (04), 831-850 (2022).

We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions N >= 6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N >= 3, for homogeneous interaction potentials with higher power.

##### Minimizers for a one-dimensional interaction energy

R. L. Frank

Nonlinear Analysis-Theory Methods & Applications 216, 112691 (2022).

We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modeling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works. (c) 2021 Elsevier Ltd. All rights reserved.

##### Reverse conformally invariant Sobolev inequalities on the sphere

R. L. Frank, T. Konig, H. L. Tang

Journal of Functional Analysis 282 (4), 109339 (2022).

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-sphere involving an operator of order 2s > n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of sand settle the question of validity of a corresponding inequality in all dimensions n >= 2. (c) 2021 Elsevier Inc. All rights reserved.

##### 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 (10), 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.

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

R. L. Frank, P. T. Nam

Calculus of Variations and Partial Differential Equations 60 (6), 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.

##### 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.

##### The Nonlinear Schrodinger Equation for Orthonormal Functions II: Application to Lieb-Thirring Inequalities

R. L. Frank, D. Gontier, M. Lewin

Communications in Mathematical Physics 384 (3), 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 Schrodinger operator -Delta + V(x) are raised to the power. is never given by the one-bound state case when kappa > max(0, 2 - d/2) in space dimension d >= 1. When in addition kappa >= 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 Schrodinger 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 Schrodinger equation admits no orthonormal ground state in 1D, for more than one function.

##### 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.

##### Equivalence of Sobolev Norms Involving Generalized Hardy Operators

R. L. Frank, K. Merz, H. Siedentop

International Mathematics Research Notices 2021 (3), 2284-2303 (2021).

We consider the fractional Schrodinger 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) Schrodinger operators and have an important application in the treatment of large relativistic atoms.

##### 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.

##### Relativistic Scott conjecture: a short proof

R.L. Frank, K. Merz, H. Siedentop

We consider heavy neutral atoms of atomic number Z modeled with kinetic energy (c^2p^2+c^4)^1/2−c^2 used already by Chandrasekhar. 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. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale.

##### 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.

##### 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.