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Elliott H. Lieb

Publications and source records attributed to Elliott H. Lieb.

At least 55 records · Page 3Linked to original sources
arXiv2014

A smooth, complex generalization of the Hobby-Rice theorem

The Hobby-Rice Theorem states that, given $n$ functions $f_j$ on ${\mathbb{R}}^N$, there exists a multiplier $h$ such that the integrals of $f_jh$ are all simultaneously zero. This multiplier takes values~$\pm1$ and is discontinuous. We show how to find a multiplier $h=e^{ig}$ that is infinitely differentiable, takes values on the unit circle, and is such that the integrals of $f_jh$ are all zero. We also show the existence of $n$ infinitely differentiable, real functions $g_j$ such that the $n$ functions $f_j e^{ig_j}$ are pairwise orthogonal.

math.FA↗
arXiv2014

Remainder Terms for Some Quantum Entropy Inequalities

We consider three von Neumann entropy inequalities: subadditivity; Pinsker's inequality for relative entropy; and the monotonicity of relative entropy. For these we state conditions for equality, and we prove some new error bounds away from equality, including an improved version of Pinsker's inequality.

quant-ph↗
arXiv2014

The entropy concept for non-equilibrium states

In earlier work we presented a foundation for the Second Law of Classical Thermodynamics in terms of the Entropy Principle. More precisely, we provided an empirically accessible axiomatic derivation of an entropy function defined on all equilibrium states of all systems that has the appropriate additivity and scaling properties and whose increase is a necessary and sufficient condition for an adiabatic process between two states to be possible. Here, after a brief review of this approach, we address the question of defining entropy for non-equilibrium states. Our conclusion is that it is generally not possible to find a unique entropy that has all relevant physical properties. We do show, however, that one can define two entropy functions, called $S_-$ and $S_+$, which, taken together, characterize the range of adiabatic processes that can occur between non-equilibrium states. The concept of {\it comparability} of states with respect to adiabatic changes plays an important role in our reasoning.

math-ph↗
arXiv2013

Upper bounds on mixing rates

We prove upper bounds on the rate, called "mixing rate", at which the von Neumann entropy of the expected density operator of a given ensemble of states changes under non-local unitary evolution. For an ensemble consisting of two states, with probabilities of p and 1-p, we prove that the mixing rate is bounded above by 4\sqrt{p(1-p)} for any Hamiltonian of norm 1. For a general ensemble of states with probabilities distributed according to a random variable X and individually evolving according to any set of bounded Hamiltonians, we conjecture that the mixing rate is bounded above by a Shannon entropy of a random variable X. For this general case we prove an upper bound that is independent of the dimension of the Hilbert space on which states in the ensemble act.

quant-ph↗
arXiv2013

Monotonicity of a relative Rényi entropy

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al.

math-ph↗
arXiv2013

Strichartz inequality for orthonormal functions

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

math.AP↗
arXiv2013

Current Densities in Density Functional Theory

It is well known that any given density rho(x)can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density rho(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x) =j(x)/rho(x) is curl free, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl free, there is a finite energy solution for all N\geq 4, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non curl free velocity field for which there is a solution, and an example for which there is no solution. The case $N=3 with a non curl free velocity field is left open.

physics.chem-ph↗
arXiv2013

Realization of stripes and slabs in two and three dimensions

We consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value J_c, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. Here we prove rigorously that the ground state energy per site converges to that of the optimal periodic striped/slabbed state, in the limit that J tends to the ferromagnetic transition point. While this theorem does not prove rigorously that the ground state is precisely striped/slabbed, it does prove that in any suitably large box the ground state is striped/slabbed with high probability.

cond-mat.stat-mech↗
arXiv2013

Stability estimates for the lowest eigenvalue of a Schrödinger operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.

math.AP↗
arXiv2013

Formation of stripes and slabs near the ferromagnetic transition

We consider Ising models in d=2 and d=3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)^(-p), p>2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J_c, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J_c the ground state is periodic and striped, with stripes of constant width h=h(J), and h tends to infinity as J tends to J_c from below. (In d=3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e_0(J)/e_S(J) tends to 1 as J tends to J_c from below, with e_S(J) being the energy per site of the optimal periodic striped/slabbed state and e_0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e_0(J)-e_S(J) at small but finite J_c-J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size l (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability.

math-ph↗
arXiv2013

Equivalence of two definitions of the effective mass of a polaron

Two definitions of the effective mass of a particle interacting with a quantum field, such as a polaron, are considered and shown to be equal in models similar to the Froehlich polaron model. These are: 1. the mass defined by the low momentum energy $E(P) \approx E(0)+P^2/2M$ of the translation invariant system constrained to have momentum $P$ and 2. the mass $M$ of a simple particle in an arbitrary slowly varying external potential, $V$, described by the nonrelativistic Schroedinger equation, whose ground state energy equals that of the combined particle/field system in a bound state in the same $V$.

math-ph↗
arXiv2013

Existence of ground states for negative ions at the binding threshold

As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition at some critical Z_c. We investigate whether the electrons remain bound when Z=Z_c and whether the radius of the system stays finite as Z_c is approached. Existence of a ground state at Z_c is shown under the condition Z_c<N-K, where K is the maximal number of electrons that can be removed at Z_c without changing the ground state energy.

math-ph↗
arXiv2013

On an Extension Problem for Density Matrices

We investigate the problem of the existence of a density matrix rho on the product of three Hilbert spaces with given marginals on the pair (1,2) and the pair (2,3). While we do not solve this problem completely we offer partial results in the form of some necessary and some sufficient conditions on the two marginals. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, trace_1 (rho_{12}) = \trace_3 (rho_{23}).

quant-ph↗
arXiv2012

Proof of an entropy conjecture for Bloch coherent spin states and its generalizations

Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum $J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from $J$ to $K=J+1/2, J+1, ...$, with $K=\infty$ corresponding to the Wehrl map to classical densities. For each $J$ and $J<K\leq \infty$ we show that the minimal output entropy for these channels occurs for a $J$ coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.

math-ph↗
arXiv2012

Ground state properties of multi-polaron systems

We summarize our recent results on the ground state energy of multi-polaron systems. In particular, we discuss stability and existence of the thermodynamic limit, and we discuss the absence of binding in the case of large Coulomb repulsion and the corresponding binding-unbinding transition. We also consider the Pekar-Tomasevich approximation to the ground state energy and we study radial symmetry of the ground state density.

math-ph↗
arXiv2012

A positive density analogue of the Lieb-Thirring inequality

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.

math-ph↗