A simple proof of Hardy-Lieb-Thirring inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).Comment: 12 page

Scott correction for large atoms and molecules in a self-generated magnetic field

We consider a large neutral molecule with total nuclear charge $Z$ in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z\al^2\le \kappa_0 for a sufficiently small $\kappa_0$, where \al denotes the fine structure constant. We show that, in the simultaneous limit $Z\to\infty$, \al\to 0 such that \kappa =Z\al^2 is fixed, the ground state energy of the system is given by a two term expansion $c_1Z^{7/3} + c_2(\kappa) Z^2 + o(Z^2)$. The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion

Extended quantum conditional entropy and quantum uncertainty inequalities

Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique of the original derivation of SSA.Comment: 4 page

Binding of Polarons and Atoms at Threshold

If the polaron coupling constant $\alpha$ is large enough, bipolarons or multi-polarons will form. When passing through the critical $\alpha_c$ from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explodes? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at $\alpha_c$. Similarly, we show that the same phenomenon occurs for atoms, e.g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schr\"odinger equation, and are very general. They use the fact that the Coulomb repulsion decays like $1/r$, while `uncertainty principle' localization energies decay more rapidly, as $1/r^2$.Comment: 19 page

Eigenvalue bounds for Schr\"odinger operators with a homogeneous magnetic field

We prove Lieb-Thirring inequalities for Schr\"odinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.Comment: 12 page

Behaviour of the energy gap near a commensurate-incommensurate transition in double layer quantum Hall systems at nu=1

The charged excitations in the system of the title are vortex-antivortex pairs in the spin-texture described in the theory by Yang et al which, in the commensurate phase, are bound together by a ``string''. It is shown that their excitation energy drops as the string lengthens as the parallel magnetic field approaches the critical value, then goes up again in the incommensurate phase. This produces a sharp downward cusp at the critical point. An alternative description based on the role of disorder in the tunnelling and which appears not to produce a minimum in the excitation energy is also discussed. It is suggested that a similar transition could also occur in compressible Fermi-liquid-like states.Comment: latex file, 17 page

Effect of smoke-free policies in outdoor areas and private places on children's tobacco smoke exposure and respiratory health:a systematic review and meta-analysis

BACKGROUND: Smoke-free policies in outdoor areas and semi-private and private places (eg, cars) might reduce the health harms caused by tobacco smoke exposure (TSE). We aimed to investigate the effect of smoke-free policies covering outdoor areas or semi-private and private places on TSE and respiratory health in children, to inform policy. METHODS: In this systematic review and meta-analysis, we searched 13 electronic databases from date of inception to Jan 29, 2021, for published studies that assessed the effects of smoke-free policies in outdoor areas or semi-private or private places on TSE, respiratory health outcomes, or both, in children. Non-randomised and randomised trials, interrupted time series, and controlled before-after studies, without restrictions to the observational period, publication date, or language, were eligible for the main analysis. Two reviewers independently extracted data, including adjusted test statistics from each study using a prespecified form, and assessed risk of bias for effect estimates from each study using the Risk of Bias in Non-Randomised Studies of Interventions tool. Primary outcomes were TSE in places covered by the policy, unplanned hospital attendance for wheezing or asthma, and unplanned hospital attendance for respiratory tract infections, in children younger than 17 years. Random-effects meta-analyses were done when at least two studies evaluated policies that regulated smoking in similar places and reported on the same outcome. This study is registered with PROSPERO, CRD42020190563. FINDINGS: We identified 5745 records and assessed 204 full-text articles for eligibility, of which 11 studies met the inclusion criteria and were included in the qualitative synthesis. Of these studies, seven fit prespecified robustness criteria as recommended by the Cochrane Effective Practice and Organization of Care group, assessing smoke-free cars (n=5), schools (n=1), and a comprehensive policy covering multiple areas (n=1). Risk of bias was low in three studies, moderate in three, and critical in one. In the meta-analysis of ten effect estimates from four studies, smoke-free car policies were associated with an immediate TSE reduction in cars (risk ratio 0·69, 95% CI 0·55-0·87; 161 466 participants); heterogeneity was substantial (I2 80·7%; p<0·0001). One additional study reported a gradual TSE decrease in cars annually. Individual studies found TSE reductions on school grounds, following a smoke-free school policy, and in hospital attendances for respiratory tract infection, following a comprehensive smoke-free policy. INTERPRETATION: Smoke-free car policies are associated with reductions in reported child TSE in cars, which could translate into respiratory health benefits. Few additional studies assessed the effect of policies regulating smoking in outdoor areas and semi-private and private places on children's TSE or health outcomes. On the basis of these findings, governments should consider including private cars in comprehensive smoke-free policies to protect child health. FUNDING: Dutch Heart Foundation, Lung Foundation Netherlands, Dutch Cancer Society, Dutch Diabetes Research Foundation, Netherlands Thrombosis Foundation, and Health Data Research UK

Uniqueness and Nondegeneracy of Ground States for (−Δ)sQ+Q−Qα+1=0(-\Delta)^s Q + Q - Q^{\alpha+1} = 0 in R\mathbb{R}

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page

Entropy and the uncertainty principle

We generalize, improve and unify theorems of Rumin, and Maassen--Uffink about classical entropies associated to quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different bases. Thus they provide a kind of uncertainty principle. Our inequalities are sharp because they are exact in the high-temperature or semi-classical limit.Comment: 6 page