arXiv subjects
Publications and source records attributed to Jan Philip Solovej.
We study the spin-J Fermi gas, interacting through a general repulsive 2-body potential, and prove asymptotics of the ground state energy in the dilute limit. The asymptotic behaviour is given in terms of the ground state energy of a spin chain, which is the Heisenberg antiferromagnet in the case of spin-1/2 fermions.
We study Berezin-Toeplitz quantization of complex projective spaces $\mathbb{CP}^{d-1}$ and obtain full asymptotic expansions of the Berezin transformation and of products of Toeplitz operators. In each case, the remainder is controlled by the next term of the expansion, either through a positivity-preserving transformation or via an operator inequality. This leads to bounds which are optimal in terms of the required regularity and feature sharp or asymptotically sharp constants.
We show that the angular momentum distribution of neutral atoms, in the sense of occupation numbers, follows the Thomas-Fermi angular momentum distribution in the limit of large atomic number. In particular, we show that the angular momentum distribution of large atoms deviates from the well-known Madelung rule.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee-Huang-Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
We study the ground state energy of a gas of 1D bosons with density $ρ$, interacting through a general, repulsive 2-body potential with scattering length $a$, in the dilute limit $ρ|a|\ll1$. The first terms in the expansion of the thermodynamic energy density are $π^2ρ^3/3(1+2ρa)$, where the leading order is the 1D free Fermi gas. This result covers the Tonks-Girardeau limit of the Lieb-Liniger model as a special case, but given the possibility that $a>0$, it also applies to potentials that differ significantly from a delta function. We include extensions to spinless fermions and 1D anyonic symmetries, and discuss an application to confined 3D gases.
We consider a Thomas-Fermi mean-field model for large neutral atoms. That is, Schrödinger operators $H_Z^{\text{TF}}=-Δ-Φ_Z^{\text{TF}}$ in three-dimensional space, where $Z$ is the nuclear charge of the atom and $Φ_Z^{\text{TF}}$ is a mean-field potential coming from the Thomas-Fermi density functional theory for atoms. For any sequence $Z_n\to\infty$ we prove that the corresponding sequence $H_{Z_n}^{\text{TF}}$ is convergent in the strong resolvent sense if and only if $D_{\text{cl}}Z_n^{1/3}$ is convergent modulo $1$ for a universal constant $D_{\text{cl}}$. This can be interpreted in terms of periodicity of large atoms. We also characterize the possible limiting operators (infinite atoms) as a periodic one-parameter family of self-adjoint extensions of $-Δ-C_\infty\vert\,x\,\vert^{-4}$ for an explicit number $C_\infty$.
We study completely positive and trace-preserving equivariant maps between operators on irreducible representations of $\mathrm{SU}(2)$. We find asymptotic approximations of channels in the limit of large output representation and we compute traces of functions of channel outputs. Our main tool is quantization using coherent states. We provide quantitative error bounds for various semiclassical formulas satisfied by quantizations of functions on the sphere.
Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large $N$-behavior of the optimal constant $κ_N$ in the Hardy inequality $\sum_{n=1}^N (-Δ_n)^s \geq κ_N \sum_{n<m} |X_n-X_m|^{-2s}$, when restricted to antisymmetric functions. We show that $N^{1-\frac{2s}d}κ_N$ 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.
We study the ground state energy of a gas of $N$ fermions confined to a unit box in $d$ dimensions. The particles interact through a 2-body potential with strength scaled in an $N$-dependent way as $N^{-α}v$, where $α\in \mathbb R$ and $v$ is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case $α<1-\frac2d$. We contrast our result with existing results in the weakly interacting case $α>1-\frac2d$, and the transition happening at the mean-field scaling $α=1-\frac2d$. Our proof is an adaptation of the bosonization technique used to treat the mean-field case.
In \cite{Nam} Nam proved a Lieb--Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with a much simplified proof. As a corollary we obtain a simple proof of the original Lieb--Thirring inequality.
For a dilute system of non-relativistic bosons interacting through a positive potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving a lower bound consistent with the Lee-Huang-Yang formula for the energy density. The proof allows for potentials with large $L^1$-norm, in particular, the case of hard core interactions is included. Thereby, we solve a problem in mathematical physics that had been a major challenge since the 1960's.
We discuss the Wehrl-type entropy inequality conjecture for the group $SU(1,1)$ and for its subgroup $AX+B$ (or affine group), their representations on $L^2({\mathbb R}_+)$, and their coherent states. For $AX+B$ the Wehrl-type conjecture for $L^p$-norms of these coherent states (also known as the Rényi entropies) is proved in the case that $p$ is an even integer. We also show how the general $AX+B$ case reduces to an unsolved problem about analytic functions on the upper half plane and the unit disc.
We extend the analysis of the Bogoliubov free energy functional to two dimensions at very low temperatures. For sufficiently weak interactions, we prove two term asymptotics for the ground state energy.
For a dilute system of non-relativistic bosons interacting through a positive, radial potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1- C \sqrt{ρa^3} \,)$.
For a dilute system of non-relativistic bosons interacting through a positive $L^1$ potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving the Lee-Huang-Yang formula for the energy density.
We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to\infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.
This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov-Bohm solenoids in Euclidean three-space, the flux carried by an oriented knot features a $2π$-periodicity of the associated operator. For a given link one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general non-trivial. In the special case of a link of unknots we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.
We study the spectral flow of Dirac operators with magnetic links on $\mathbb{S}^3$. These are generalisations of Aharonov-Bohm solenoids where the magnetic fields contain finitely many field lines coinciding with the components of a link, the flux of each exhibiting the same $2π$-periodicity as A-B solenoids. We study the spectral flow of the loop obtained as tuning the flux from $0$ to $2π$ in the case of only one field line: we relate the spectral flows obtained for one given knot and its cable knots, and obtain that torus knots have trivial spectral flow. The operators are studied in their Coulomb gauge in $\mathbb{R}^3$ (seen as a chart of $\mathbb{S}^3$ through the stereographic projection), which is simply given by the Biot and Savart formula.