arXiv subjects
Publications and source records attributed to Robert Seiringer.
We give a CLR type bound on the number of bound states of Schroedinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive $δ$-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length $a$ and the radius $r$ of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size $\sim r^{-2}$ above the ground state energy.
It was recently shown by Hansen that the Wigner-Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting applications in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample.
After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein Condensation is presented. The proof is based on a rigorous validation of Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator $a_{\bf 0}$.
We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schroedinger-like operators remain true, with possibly different constants, when the critical Hardy-weight $C|x|^{-2}$ is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schroedinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge $Zα=2/π$.
We give a proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge $Zα=2/π$.
This book surveys results about the quantum mechanical many-body problem of the Bose gas that have been obtained by the authors over the last seven years. These topics are relevant to current experiments on ultra-cold gases; they are also mathematically rigorous, using many analytic techniques developed over the years to handle such problems. Some of the topics treated are the ground state energy, the Gross-Pitaevskii equation, Bose-Einstein condensation, superfluidity, one-dimensional gases, and rotating gases. The book also provides a pedagogical entry into the field for graduate students and researchers.
A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $ρ$ and temperature $T$. In the dilute regime, i.e., when $a^3ρ\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term $4πa (2ρ^2 - [ρ-ρ_c]_+^2)$. Here, $ρ_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., $T \sim ρ^{2/3}$ or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [arXiv:math-ph/0601051] for estimating correlations to temperatures below the critical one.
Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.
We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation.
We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of $N$ fermions interacting with a short-range, positive potential of scattering length $a$. For spin 1/2 fermions, this is $E \sim E^0 + (\hbar^2/2m) 2 πN ρa$, where $E^0$ is the energy of the non-interacting system and $ρ$ is the density.
The validity of substituting a c-number $z$ for the $k=0$ mode operator $a_0$ is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities like the pressure or ground state energy, but also the value of $|z|^2$ that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry breaking term -- a point that had previously been elusive.
We consider a gas of fermions with non-zero spin at temperature $T$ and chemical potential $μ$. We show that if the range of the interparticle interaction is small compared to the mean particle distance, the thermodynamic pressure differs to leading order from the corresponding expression for non-interacting particles by a term proportional to the scattering length of the interparticle interaction. This is true for any repulsive interaction, including hard cores. The result is uniform in the temperature as long as $T$ is of the same order as the Fermi temperature, or smaller.
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of $N$ fermions interacting with a short-range, positive potential of scattering length $a$. For spin 1/2 fermions, this is $E \sim E^0 + (\hbar^2/2m) 2 πN ρa$, where $E^0$ is the energy of the non-interacting system and $ρ$ is the density. A similar formula holds in 2D, with $ρa$ replaced by $ρ/|\ln(ρa^2)|$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[ρ]= - Tr (ρ\ln ρ) of a density matrix ρ_{123} on the product of three Hilbert spaces satisfies S[ρ_{123}] - S[ρ_{23}] \leq S[ρ_{12}]- S[ρ_2]. We strengthen this to S[ρ_{123}] - S[ρ_{12}] \leq \sum_αn^α(S[ρ_{23}^α] - S[ρ_2^α]), where the n^αare weights and the ρ_{23}^αare partitions of ρ_{23}. Correspondingly, there is a strengthening of the theorem that the map A -> Tr \exp[L + \ln A] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl entropy and entropy inequalities for quantum gases.
This article gives a detailed presentation of the authors' recent results on the ground state properties of the Bose gas. It is a much expanded version of a talk given by one of the authors (E.H.L.) at the conference "Perspectives in Analysis" at the KTH, Stockholm, June 23, 2003. It is based on, but supersedes, the article math-ph/0204027.
The classic Poincare inequality bounds the $L^q$-norm of a function $f$ in a bounded domain $Ω\subset \R^n$ in terms of some $L^p$-norm of its gradient in $Ω$. We generalize this in two ways: In the first generalization we remove a set $Γ$ from $Ω$ and concentrate our attention on $Λ= Ω\setminus Γ$. This new domain might not even be connected and hence no Poincare inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of $Γ$ is arbitrarily small. A Poincare inequality does hold, however, if one makes the additional assumption that $f$ has a finite $L^p$ gradient norm on the whole of $Ω$, not just on $Λ$. The important point is that the Poincare inequality thus obtained bounds the $L^q$-norm of $f$ in terms of the $L^p$ gradient norm on $Λ$ (not $Ω$) plus an additional term that goes to zero as the volume of $Γ$ goes to zero. This error term depends on $Γ$ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the `nice' domain $Ω$. In the second generalization we are given a vector field $A$ and replace $\nabla $ by $\nabla +i A(x)$ (geometrically, a connection on a U(1) bundle). Unlike the A=0 case, the infimum of $\|(\nabla +i A)f\|_p $ over all $f$ with a given $\|f\|_q$ is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.