Inequalities

The subject of ‘inequalities’ was first systematically established by Hardy, Littlewood and P61ya in their book of the same name. The goal, loosely speaking, is to search for an inequality between algebraic or analytic expressions of certain variables that becomes an equality in certain (possibly limiting) cases. The reader may think of Holder’s inequality as an example, but also such things as the dependence upon its shape of the lowest frequency of a drum.

Consider a system of N electrons in one dimension subject to an arbitrary symmetric potential, V(x₁…,XN), and let E(S) be the lowest energy belonging to the total spin value S. We have proved the following theorem: E(S) <E(S’) if S>Sr. Hence, the ground state is unmagnetized. The theorem also holds in two or three dimensions (although it is possible to have degeneracies) provided V(x ₁ ,y ₁ ,Z ₁ ; …; x _n ,yn, Z _N ) is separately symmetric in the x_iy_i and Z_i. The potential need not be separable, however. Our theorem has strong implications in the theory of ferromagnetism because it is generally assumed that for certain repulsive potentials, the ground state is magnetized. If such be the case, it is a very delicate matter, for a plausible theory must not be so general as to give ferromagnetism in one dimension, nor in three dimensions with a separately symmetric potential

The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetie Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S) ] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S ≥ 0. In ferrimagnetism, E(S + 1) > E(S) for S ≥ S, and in addition the ground state belongs to S ≤ S. S is defined as follows: Let the maximum spin of the A sublattice be S_A and of the B sublattice S_B; then S = S_A—S_B. Antiferromagnetism is treated as the special case of S = 0. We also briefly discuss the structure of the lowest eigenfunctions in an external magnetic field.

Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence of the limiting mean entropy for translationally invariant states of quantum continuous systems.

There are some properties of entropy, such as concavity and subadditivity, that are known to hold (in classical and in quantum mechanics) irrespective of any assumptions on the detailed dynamics of a system. These properties are consequences of the definition of entropy as S(p) =—Trp lnp (quantum), (1a) S(p) =- f_p lnp (classical continuous), (1b) S(p)= p_i Inp_i (classical discrete), (1c) where Tr means trace, p is a density matrix in (1a), and p is a distribution function (usually on R6n) in (1b). In (1c) the p_i are discrete energy level probabilities.

In this paper we prove several theorems about quantum mechanical entropy, in particular, that it is strongly subadditive (SSA). These theorems were announced in an earlier note,1 to which we refer the reader for a discussion of the physical significance of SSA and for a review of the historical background. We repeat here a bibliography of relevant papers.2-9.

Introduction. Statistical mechanics is the science of explaining, predicting and understanding the gross, macroscopic attributes of matter (which may be taken to mean mechanical systems with essentially an infinite number of degrees of freedom) in terms of the elementary dynamical laws governing its atomic constituents. The problems that arise are sufficiently complex and intriguing, but at the same time sufficiently well posed, that the subject is nowadays as much a part of mathematics as of physics. The fields of information theory and ergodic theory had their genesis in statistical mechanical modes of thought and are now well established in the mathematics literature; there will be more to come.

A general formulation is given of Simon’s Ising model inequality : $$ \left\langle {{\sigma _\alpha }{\sigma _\gamma }} \right\rangle \le \sum\limits_{b \in B} {\left\langle {{\sigma _\alpha }{\sigma _b}} \right\rangle \left\langle {{\sigma _b}{\sigma _\gamma }} \right\rangle }$$ Where B is any set of spins separating a from δ. We show that (σ{α}σ_b) can be replaced by (σ_{ασb)A where A is the spin system “inside” B containing α. An advantage of this is that a finite algorithm can be given to compute the transition temperature to any desired accuracy. The analogous inequality for plane rotors is shown to hold if a certain conjecture can be proved. This conjecture is indeed verified in the simplest case, and leads to an upper bound on the critical temperature. (The conjecture has been proved in general by Rivasseau. See notes added in proof.}

In the attractive Hubbard model (and some extended versions of it), the ground state is proved to have spin angular momentum S = 0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S = 1/2 || B |—| A ||, where |B| ( |A| ) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the |B| = |A| case and yields, with | B | ≠ | A|, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.

The Lieb-Mattis theorem on the absence of one-dimensional ferromagnetism is extended here from ground states to T> 0 by proving, inter alia, that M(ß,h), the magnetization of a quantum system in a field h> 0, is always less than the pure paramagnetic value M_o(ß,h)=lanh(ßh), with ß=1/kT. Our proof rests on a new formulation in terms of path integrals that holds in any dimension; another of its applications is that the Nagaoka-Thouless theorem on the Hubbard model also extends to T > 0 in the sense that M(ß,h ) exceedsM₀(ß,h ).

In a recent paper [1] Marcus and Minc have given an admirable summary of the theory of permanents of square matrices, especially with regard to inequalities satisfied by the permanent. Our knowledge of permanents is certainly meager compared to that for determinants, largely for the reason that while the former differs from the latter only in the replacement of minus signs by plus signs, the determinant, and not the permanent, is invariant under unitary transformations.

The Stirling numbers of the first kind, S_Nk, and of the second kind, σ_Nk, are shown to be strongly logarithmically concave as functions of k for fixed TV. This result is stronger than the unimodality conjecture which was heretofore proved only for σ_Nk (Harper). We also introduce a generating function for the σ_Nk which is different from the conventional one but which has a relatively simple closed form expression.

Several convex mappings of linear operators on a Hilbert space into the real numbers are derived, an example being A → — Tr exp(L + In A). Some of these have applications to physics, specifically to the Wigner—Yanase—Dyson conjecture which is proved here and to the strong subadditivity of quantum mechanical entropy which will be proved elsewhere.

We prove several operator inequalities which are analogous to the Schwarz inequality.

In this note we generalize an inequality on determinants (Corollary 3 below) recently proved by Seiler and Simon [1] in connection with some estimates in quantum field theory. Our main result is Lemma 1.

When studying the third law of thermodynamics for an infinite system, or properties of equilibrium states of such a system (see [3]), one is led to consider the effect of local perturbations.

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established.

We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace ideals C_p, which are the analogs of the Lebesgue spaces L_p in non-commutative integration. The inequalities are all precise analogs of results which had been known in L_p, but were only known in C_p for special values of p. In the course of our treatment of uniform convexity and smoothness inequalities for C_p we obtain new and simple proofs of the known inequalities for L_p.

We consider the following trace function on n-tuples of positive operators: $${\Phi _P}({A_1},{A_2},...,{A_n}) = Tr({(\sum\limits_{j = 1}^n {A_j^P} )^{1/P}})$$ and prove that it is jointly concave for 0 < p ≤ 1 and convex for p = 2. We then derive from this a Minkowski type inequality for operators on a tensor product of three Hilbert spaces, and show how this implies the strong subadditivity of quantum mechanical entropy. For p > 2, Ф_p is neither convex nor concave. We conjecture that Ф_p is convex for 1 < p < 2, but our methods do not show this.

Estimates for the number of bound states and their energies, e_j ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = ≤ Δ + V(x) in Rn, we shall use available methods to derive the bounds 1$${\sum\limits_j {\left| {{e_j}} \right|} ^y} \le {L_{y,n}}\int {{d^n}} x\left| {V(x)} \right|_ - ^{y + n/2},y > \max (0,1 - n/2)$$ Here, $$\left| {V(x)} \right|\_ = - V(x){\rm{ if V(x)}} \le {\rm{0}}$$ and is zero otherwise.

Our principal result is that if the semiclassical estimate is a bound for some moment of the negative eigenvalues (as is known in some cases in one-dimension), then the semiclassical estimates are also bounds for all higher moments.

If N ((Ω,λ) is the number of eigenvalues of -Δ in a domain Ω, in a suitable Riemannian manifold of dimension n, we derive bounds of the form $$\tilde N(\Omega ,\lambda ) \le {D_n}{\lambda ^{n/2}}\left| \Omega \right|$$ for all Ω, * , n , Likewise, if N_03B1; (V) is the number of nonpositive eigenvalues of -Δ + V (x) which are ≤ a ≤ 0, then $${N_\alpha }(V) \le {L_n}\int {_M} \left[ {V - \alpha } \right]_\_^{n/2}$$ for all α and V and n ≥ 3. 1980 Mathematics Subject Classification 35P15.

For a Coulomb system of particles of charge e, it has previously been shown that the indirect part of the repulsive Coulomb energy (exchange plus correlation energy) has a lower bound of the form -Ce2/3fp(x)4/3dx, where p is the single particle charge density. Here we lower the constant C from the 8.52 previously given to 1.68. We also show that the best possible C is greater than 1.23.

This paper has three aims: (i) To discuss some of the mathematical connections between N-particle wave functions Ψ and their single-particle densities p(x). (ii) To establish some of the mathematical underpinnings of “universal density functional” theory for the ground state energy as begun by Hohenberg and Kohn. We show that the HK functional is not defined for all p and we present several ways around this difficulty. Several less obvious problems remain, however, (iii) Since the functional mentioned above is not computable, we review examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy.

Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Ω. The latter is particularly important because it yields an upper bound to the Hausdorff dimension of attracting sets. However, Ruelle’s bound on the number has three deficiences : (i) it relies on some unproved conjectures about certain constants; (ii) it is valid only in dimensions ≧ 3 and not 2 ; (iii) it is valid only in the limit Ω-→ ∞. In this paper these deficiences are remedied and, in addition, the final constants in the inequality are improved.

Recently conjectured three- (and more-) body mass inequalities are investigated for the quark models of baryons in which it is assumed that baryon masses are the ground-state energies of Schrödinger-type operators with pair potentials V. It is proved that these inequalities hold (even with a “relativistic” form for the kinetic energy) if V belongs to a certain class (which includes many potentials commonly used), but that they do not hold for all V (even in the nonrelativistic case). One example of our results is 2M(cqs)≥ M(cqq) + M(css).

he Sobolev inequality on Rn,n ≥ 3 is very important because it gives a lower bound for the kinetic energy f| ∇f|2 in terms of an Lp norm of f.

We give a proof of the Lieb-Thirring inequality in the critical case d=1, γ = 1/2, which yields the best possible constant.

We derive a classical integral representation for the partition function, ZQ, of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentum J→∞ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality is ZC(J) ≦ ZQ(J) ≦ZC(J +1).

Wehrl has proposed a new definition of classical entropy, S, in terms of coherent states and conjectured that s≧ 1. A proof of this is given. We discuss the analogous problem for Bloch coherent spin states, but in this case the conjecture is still open. An inequality for the entropy of convolutions is also given.

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.

This talk reviews some of the ways in which coherent states can be used to give rigorous bounds for quantities of physical interest and, in certain cases, can yield exact asymptotic formulas. Three main topics will be discussed.

In this paper we prove a rearrangement inequality that generalizes inequalities given in the b∞k by Hardy, Littlewood and Pólya1 and by Luttinger and Friedberg.2 The inequality for an integral of a product of functions of one variable is further extended to the case of functions of several variables.

THE following is a preliminary report on some recent work, the full details of which will be published elsewhere. We have come across some inequalities about integrals and moments of log concave functions which hold in the multidimensional case and which are useful in obtaining estimates for multidimensional modified Gaussian measures. By making a small jump (we shall not go into the technical details) from the finite to the infinite dimensional case, upper and lower bounds to certain types of functional integrals can be obtained. As a non-trivial application of the latter we shall, for the first time, prove that the one-dimensional one-component quantummechanical plasma has long-range order when the interaction is strong enough. In other words, the Wigner lattice can exist, in one dimension at least. As another application we shall prove a log concavity theorem about the fundamental solution (Green’s function) of the diffusion equation.

The best possible constant Dmt in the inequality | ∬ dx dyf(x)g(x —y) h(y)|< D_pgt||f||p||g||Q||h||t, p,q,t>|, 1/p + llq+ 1/t = 2, is determined; the equality is reached if /, g, and h are appropriate Gaussians. The same is shown to be true for the converse inequality (0 < p, q < 1, t < 0), in which case the inequality is reversed. Furthermore, an analogous property is proved for an integral of k functions over n variables, each function depending on a linear combination of the n variables; some of the functions may be taken to be fixed Gaussians. Two applications are given, one of which is a pr∞f of Nelson’s hypercontractive inequality.

We extend the Prékopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prékopa—Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prékopa—Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that & is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetric decreasing rearrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing Φ. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.

Suppose f(xl,x2) ≥ 0 is a continuously differentiable function supported in the unit disk in the plane. Its symmetric decreasing rearrangement is the rotationally invariant function f*(xl,x2) whose level sets are circles enclosing the same area as the level sets of f. Such rearrangement preserves L_p norms but decreases convex gradient integrals, e.g. ||∇||*||_p ≤ ||∇/||_p (1 ≤ p < ∞). Now suppose that fj(x1,x2) > 0 (j = 1,2,3,…) is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f. The symmetzed functions also converge uniformly. The real question is about convergence of the derivatives of the symmetrized functions. We announce that the derivatives of the symmetrized functions need not converge strongly, e.g. it can happen that ||∇fj*—∇f*||p →* 0 for every p. We further characterize exactly those f’s for which convergence is assured and for which it can fail

The operation R, of symmetric decreasing rearrangement maps W1,P( Rn) to W1,p( Rn). Even though it is norm decreasing we show that R is not continuous for n>2. The functions at which R is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiate function is co-area regular, and both the regular and the irregular functions are dense in W1 ,p(Rn).

Suppose that Ω and Ω₁ are convex, open subsets of Rn. Denote their convex combination by The Brunn-Minkowski inequality says that (vol Ω)_t≥ (1 -t) vol Ω₀1/N +t Vol Ω for 0≤t ≤ l. Moreover, if there is equality for some t other than an endpoint, then the domains Ω₁ and Ω₀ are translates and dilates of each other. Borell proved an analogue of the Brunn—Minkowski inequality with capacity (defined below) in place of volume. Borel’s theorem [B] says THEOREM A. Let Ωt= tΩ1+ (1—t)Ω0 be a convex combination of two convex subsets of RN,N≥3. Then cap The main purpose of this note is to prove.

Let $${\Psi _1},...,{\Psi _N}$$be orthonormal functions in Rd and let $${u_1} = {( - \Delta )^{ - 1/2}}{\Psi _i}$$ or $${u_1} = {( - \Delta + 1)^{ - 1/2}}{\Psi _i}$$ and let $$p(x) = {\sum {\left| {{u_i}(x)} \right|} ^2}$$. Lp bounds are proved for p, an example being for $${\left\| P \right\|_P} \le {A_d}{N^{1/p}}for{\rm{ d}} \ge {\rm{3, with p = d(d - 2}}{{\rm{)}}^{ - 1}}$$. The unusual feature of these bounds is that the orthogonality of the ψ_i yields a factor N1/P instead of N, as would be the case without orthogonality. These bounds prove some conjectures of Battle and Federbush (a Phase Cell Cluster Expansion for Euclidean Field Theories, I, 1982, preprint) and of Conlon (Comm. Math. Phys., in press).

We show that if f_n is a sequence of uniformly Lp-bounded functions on a measure space, and if f_n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to some functional other than the L_p norm, namely → 0 for suitable j: C → C and a suitable sequence f_n. A brief discussion is given of the usefulness of this result in variational problems.

A maximizing function, f, is shown to exist for the HLS inequality on Rn

If A and B are two bounded domains in Rn and λ(A) γ(B) are the lowest eigenvalues of — Δ with Dirichlet boundary conditions, then there is some translate, B_x, of B such that λ(A B_x)< λ(A) + λ(B). A similar inequality holds for. There are two corollaries of this theorem: (i) A lower bound for sup_x volume (A B_x) in terms of λ(A), when B is a ball; (ii) A compactness lemma for certain sequences in W1,p(Rn).

The system of equations studied in this paper is— Δu_i = g_i(u) on Rd, d≧ 2, with u:Rd-→]Rn and gi(u)=∂G/∂_i. Associated with this system is the action, S(u) = f {1/2|2 — G(u)}. Under appropriate conditions on G (which differ for d = 2 and d≧ 3) it is proved that the system has a solution, u 0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, u0}.

The usual Sobolev inequality in ℝn, n≥3, asserts that 1$$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2$$, with S_n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ℝn. Two kinds of inequalities are established: (i) If f = 0 on ∂ Ω, then 2$$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2 + C(\Omega )\left\| f \right\|_{p,w}^2$$ with p=2*/2. Some further results and open problems in this area are also presented.

A Gaussian integral kernel G(x, y) on Rn x Rn is the exponential of a quadratic form in x and y; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound of G as an operator from Lp(Rp) to Lq(Rn) and to prove that the LP(Rn) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1 < p ≦ q < ∞ and also for p > q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young’s inequality.

An upper bound is proved for the Lp norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p >2. A lower bound is proved for 1 ≤p < 2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the LP bounds if and only if the functions/ and g that enter the definition are both Gaussians.

Two problems concerning maps cp with point singularities from a domain} Ω = R3 to S2are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ = g is given on ΦΩ; we show that the only cases in which g (x/|x|) minimizes the energy is g = const, or g(x)= ±R x with R a rotation

We study maps ϕ from the unit ball B in R3 to the unit sphere S2 in R3 which minimize Dirichlet’s energy integral $$ \varepsilon (\varphi ) = \int {_B} |\nabla \varphi {|^2}dV $$.

Oriented n area minimizing surfaces (integral currents) in Mm+n can be approximated by level sets (slices) of nearly m-energy minimizing mappings Mm+n → Sm with essential but controlled discontinuities. This gives new perspective on multiplicity, regularity, and computation questions in least area surface theory.

Energy minimizing harmonic maps from the ball to the sphere arise in the study of liquid crystal geometries and in the classical nonhnear sigma model. We linearly dominate the number of points of discontinuity of such a map by the energy of its boundary value function. Our bound is optimal (modulo the best constant) and is the first bound of its kind. We also show that the locations and numbers of singular points of minimizing maps is often counterintuitive; in particular, boundary symmetries need not be respected.

The Ginzburg-Landau energy minimization problem for a vector field on a two dimensional disc is analyzed. This is the simplest nontrivial example of a vector field minimization problem and the goal is to show that the energy minimizer has the full geometric symmetry of the problem. The standard methods that are useful for similar problems involving real valued functions cannot be applied to this situation. Our main result is that the minimizer in the class of symmetric fields is stable, i.e., the eigenvalues of the second variation operator are all nonnegative.