A Proof of the Lieb–Thirring Inequality via the Besicovitch Covering Lemma

Abstract

We give a proof of the Lieb–Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma.

1 Introduction

The celebrated Lieb–Thirring inequality [16, 17] states that for every trace class operator 0 ≤ γ ≤ 1 on $L^{2}(\mathbb R^{d})$ with density ρ_γ(x) = γ(x,x), the kinetic energy lower bound

$$ \text{Tr}(-{\varDelta} \gamma) \ge K_{d} {\int}_{\mathbb R^{d}} \rho_{\gamma}(x)^{1+\frac{2}{d}} \mathrm{d} x $$

(1)

holds with a constant K_d > 0 depending only on the dimension d ≥ 1 (in particular, K_d is independent of γ). Here assuming the spectral decomposition

$$ \gamma= \sum\limits_{n=1}^{\infty} \lambda_{n} |u_{n}\rangle \langle u_{n}| $$

(2)

with $\{u_{n}\}_{n=1}^{\infty }$ being orthonormal in $L^{2}(\mathbb R^{d})$ and 0 ≤ λ_n ≤ 1, we denote

$$ \text{Tr}(-{\varDelta} \gamma) = \sum\limits_{n=1}^{\infty} \lambda_{n} {\int}_{\mathbb R^{d}} |\nabla u_{n}|^{2}, \quad \rho_{\gamma}(x)=\sum\limits_{n=1}^{\infty} \lambda_{n} |u_{n}(x)|^{2}. $$

(3)

Strictly speaking, the trace class condition $\text {Tr} \gamma ={\sum }_{n=1}^{\infty } \lambda _{n} <\infty$ is not essential. The bound (1) still holds for every operator 0 ≤ γ ≤ 1 provided that γ admits a spectral decomposition as in (2) such that we can define the kinetic energy Tr(−Δγ) and the density ρ_γ via (3) (this extension can be obtained easily from a density argument using the Lebesgue monotone convergence theorem). In connection to quantum physics, γ usually stands for the one-body density matrix of a quantum state, and the trace class condition means that the number of particles is finite. A key feature of (1) is that the bound is independent of the number of particles, thus allowing us to treat large systems (including even infinite systems via a standard limiting argument).

By some sort of the uncertainty principle (e.g., Sobolev’s inequality), it is not difficult to see that

$$ \left( {\int}_{\mathbb R^{d}} |\nabla u|^{2} \right) \left( {\int}_{\mathbb R^{d}} |u|^{2} \right)^{\frac 2 d} \ge K_{\text{So}} {\int}_{\mathbb R^{d}} |u(x)|^{2(1+\frac{2}{d})} \mathrm{d} x, \quad \forall u\in H^{1}(\mathbb R^{d}). $$

(4)

Here the optimal constant K_So > 0 depends only on the dimension d ≥ 1. Consequently, if $\{u_{n}\}_{n=1}^{N}$ is a family of normalized functions in $L^{2}(\mathbb R^{d})$, then using (4), we have

$$ \sum\limits_{n=1}^{N} {\int}_{\mathbb R^{d}} |\nabla u_{n}|^{2} \ge K_{\text{So}} {\int}_{\mathbb R^{d}} \sum\limits_{n=1}^{N} |u_{n}|^{2(1+\frac{2}{d})} \ge \frac{K_{\text{So}} }{N^{2/d}} {\int}_{\mathbb R^{d}} \left( \sum\limits_{n=1}^{N} |u_{n}|^{2} \right)^{1+\frac 2 d}. $$

(5)

The optimality of the constant K_SoN^− 2/d in (5) can be seen easily by considering the special case u_n = u for all n. On the other hand, if $\{u_{n}\}_{n=1}^{N}$ are orthonormal, then the Lieb–Thirring inequality (1) with $\gamma ={\sum }_{n=1}^{N} |u_{n}\rangle \langle u_{n}|$ implies that (5) holds with K_SoN^− 2/d replaced by a constant K_d depending only on the dimension d ≥ 1, which is very useful when $N\to \infty$. The orthogonality here essentially corresponds to the condition 0 ≤ γ ≤ 1 in (1).

As an application, the Lieb–Thirring inequality (1) gives a semiclassical estimate for the kinetic energy of fermionic systems, which is a crucial ingredient of the proof of the stability of matter in [16]. In this aspect, the constraint 0 ≤ γ ≤ 1 corresponds to Pauli’s exclusion principle of fermions (see, e.g., [15, Theorem 3.2]). Therefore, (1) can be interpreted as an elegant combination of the uncertainty and exclusion principles in quantum mechanics.

Historically, in the original proof of Lieb and Thirring in [16, 17], the kinetic bound (1) was derived from its dual version

$$ \text{Tr}[(-{\varDelta} + V(x))_{-}] \ge -L_{d} {\int}_{\mathbb R^{d}} |V_{-}(x)|^{1+\frac d 2} \mathrm{d} x, $$

(6)

where $t_{-}=\min \limits (t,0)$ and

$$ K_{d}\left( 1 + \frac 2 d \right) = \left[ L_{d} \left( 1+ \frac d 2\right) \right]^{-\frac 2 d}. $$

In this approach, the sum of all negative eigenvalues of the Schrödinger operator −Δ + V (x) is estimated via the number of eigenvalues ≥ 1 of the Birman–Schwinger operator $\sqrt {|V_{-}(x)|} (-{\varDelta } +E)^{-1} \sqrt {|V_{-}(x)|}$ with E > 0. This duality approach is powerful since it relates the nonlinear inequality (1) to a more linear problem, namely a spectral property of a linear operator. The spectral estimate (6) is interesting in its own right and has been generalized in various directions (see [10] for a recent review with many open questions).

In this article, we will give a direct proof of the kinetic bound (1), without relying on the duality argument. The first proof in this direction goes back to the work of Eden and Foias in 1991 [9] for one dimension, which was later extended by Dolbeault, Laptev, and Loss [6] for higher dimensions. More recently, two other direct proofs were given by Rumin [25, 26] and by Lundholm and Solovej [19, 20]; we refer to [23] for a review on these two approaches (see also [27] for another direct proof).

We will discuss a simplification of the Lundholm–Solovej approach. The key observation of this approach, which actually goes back to the first proof of the stability of matter by Dyson and Lenard [7], is that a local version of the Lieb–Thirring inequality on cubes can be obtained easily by local versions of the uncertainty and exclusion principles. As realized in [19, 20], it is possible to obtain the global bound (1) by combining such local estimates by choosing an appropriate family of disjoint cubes. A simplification of the Lundholm–Solovej approach was already given by Lundholm–Nam–Portmann [18] where the local uncertainty and exclusion principles are combined by using a new covering lemma which is inspired by a “stopping time argument” in harmonic analysis. Our aim here is to provide with a further simplification where the local uncertainty and exclusion principles are combined naturally via the classical Besicovitch covering lemma, thus making the proof very elementary. We hope that our proof will be not only interesting from the pedagogical point of view, but also helpful for finding new generalizations of (1) in the future. In particular, we expect that the method represented here can be strengthened by coupling with the ideas of using dyadic sets in recent works [12, 21].

Note that the Besicovitch covering lemma was already used in Rozenblum’s proof of the Cwikel–Lieb–Rozenblum inequality [5, 13, 24] on the number of negative eigenvalues of Schrödinger operators for d ≥ 3

(7)

In this approach, using Sobolev’s inequality, the linear operator −Δ + V (x) can be localized in bounded domains such that it has at most one bound state in each domain, and then the number of the domains is controlled by the Besicovitch covering lemma. Heuristically, since (7) is stronger than (6) (when d ≥ 3), it is not surprising that the Besicovitch covering lemma is also helpful for the proof of the Lieb–Thirring inequality. However, since (1) seems less linear than (6) and (7), we think it is still interesting to see that a variant of this microlocal technique works directly for (1).

2 Proof

Our proof of (1) includes two steps. In the first step, we derive a local version of (1) in any ball $B\subset \mathbb R^{d}$ such that ${\int \limits }_{B} \rho _{\gamma } =2$ (we can replace 2 by any fixed number bigger than 1). This step is done by combining local versions of the uncertainty and exclusion principles, exactly in the spirit of the proofs in [18,19,20] (the only difference is that we are interested in balls rather than cubes). In the second step, we cover $\mathbb R^{d}$ by a suitable collection of balls via the Besicovitch covering lemma, and conclude (1).

Now we come to the details. Let us denote, with the spectral decomposition $\gamma = {\sum }_{n} \lambda _{n} |u_{n}\rangle \langle u_{n}|$ and any ball $B\subset \mathbb R^{d}$, that

$$ \text{Tr}(-{\varDelta}_{B} \gamma) = \sum\limits_{n} \lambda_{n} {\int}_{B} |\nabla u_{n}|^{2}, \quad \rho_{\gamma}(x)=\sum\limits_{n} \lambda_{n} |u_{n}(x)|^{2}. $$

(8)

We will always denote by C_d > 0 a general (large) constant depending only on the dimension d ≥ 1. The value of C_d may change line by line.

Step 1: First, we prove a local version of (1).

Lemma 1

(Lieb–Thirring inequality for balls) For any ball $B\subset \mathbb R^{d}$ and any trace class operator 0 ≤ γ ≤ 1 on L²(B) satisfying ${\int \limits }_{B} \rho _{\gamma } = 2$, we have

$$ \text{Tr}(-{\varDelta}_{B} \gamma) \ge \frac{1}{C_{d}} {\int}_{B} \rho_{\gamma}(x)^{1+\frac{2}{d}} \mathrm{d} x. $$

The constraint ${\int \limits }_{B} \rho _{\gamma } = 2$ can be replaced by $a\le {\int \limits }_{B} \rho _{\gamma } \le b$ for any constants $1<a<b<\infty$ (then C_d depends on d,a,b).

Lemma 1 is a consequence of the following well-known results.

Lemma 2

(Hoffmann-Ostenhof inequality) For any measurable subset $B\subset \mathbb R^{d}$ and any trace class operator 0 ≤ γ ≤ 1 on L²(B) we have

$$ \text{Tr}(-{\varDelta}_{B} \gamma) \ge {\int}_{B} |\nabla \sqrt{\rho_{\gamma}}|^{2}. $$

Proof

This result was first proved in [11]. For the reader’s convenience, here we recall another derivation using the diamagnetic inequality |∇|u|||≤|∇u| (see [14, Theorem 7.21]). Indeed, for two complex functions $\{v_{n}\}_{n=1}^{2}$, using the diamagnetic inequality for v₁,v₂ and |v₁| + i|v₂| (with i² = − 1), we have

$$ |\nabla v_{1}|^{2} + |\nabla v_{2}|^{2} \ge |\nabla |v_{1}||^{2} + |\nabla |v_{2}||^{2} = |\nabla (|v_{1}| + \textbf{i} |v_{2}|)|^{2} \ge \left |\nabla \sqrt{ |v_{1}|^{2} + |v_{2}|^{2}}\right|^{2}. $$

(9)

Note that (9) is called the convexity of the gradient which was first published in [2, Theorem 4.3] and [3, Lemma 4] (see also [14, Theorem 7.8]). By induction, we find that for any functions $\{v_{n}\}_{n=1}^{\infty }$,

$$ \sum\limits_{n=1}^{\infty} |\nabla v_{n}|^{2} \ge \left|\nabla \sqrt{ \sum\limits_{n=1}^{\infty} |v_{n}|^{2}}\right|^{2}. $$

(10)

We obtain the desired estimate by integrating the pointwise inequality (10) with $v_{n}=\sqrt {\lambda _{n}} u_{n}$ and using the definition (8). □

Lemma 3

(Local uncertainty principle) For any ball $B\subset \mathbb R^{d}$ and any function g ∈ H¹(B), we have

$$ {\int}_{B} |\nabla g|^{2} \ge \frac{1}{C_{d}} \frac{{\int}_{B} |g|^{2(1+ \frac 2 d)}}{({\int}_{B} |g|^{2})^{\frac 2 d}} - \frac{C_{d}}{ |B|^{\frac 2 d}} {\int}_{B} |g|^{2} . $$

Proof

By scaling, we may assume that B is the unit ball. Then by a standard result (see [1, Theorem 7.41]), we can find $G\in H^{1}(\mathbb R^{d})$, an extension of g, such that

$$G_{|B}=g,\quad \|G\|_{H^{1}(\mathbb R^{d})} \le C_{d} \|g\|_{H^{1}(B)},\quad \|G\|_{L^{2}(\mathbb R^{d})} \le C_{d} \|g\|_{L^{2}(B)}.$$

Thus it suffices to show the Gagliardo-Nirenberg type inequality

$$ \|G\|_{L^{2(1+2/d)}(\mathbb R^{d})} \le C_{d} \|G\|_{H^{1}(\mathbb R^{d})}^{\frac d {d+2}} \|G\|_{L^{2}(\mathbb R^{d})}^{\frac 2 {d+2}},\quad \forall G\in H^{1}(\mathbb R^{d}), $$

(11)

which is indeed equivalent to Sobolev’s inequality $\|G\|_{L^{2(1+2/d)}} \le C_{d} \|G\|_{H^{1}}$ (Sobolev’s inequality seems weaker, but we can apply it for G_ℓ(x) = ℓ^d/2G(ℓx) and then optimize over ℓ > 0 to obtain (11)) (see also [18, Lemma 8] for another derivation of (11) via the fractional Sobolev embedding theorem). □

Lemma 4

(Local exclusion principle) For any open ball $B\subset \mathbb R^{d}$ and any trace class operator 0 ≤ γ ≤ 1 on L²(B)

$$ \text{Tr}(-{\varDelta}_{B} \gamma) \ge \frac{1}{C_{d} |B|^{2/d}} \Big({\int}_{B} \rho_{\gamma} - 1\Big). $$

(12)

Proof

This bound goes back to Dyson and Lenard [7]. It is a consequence of the Neumann spectral gap

where , the rank-one projection on the unique ground state of the Neumann Laplacian. Then we can take the trace against γ and conclude by using and Tr(Pγ) ≤TrP = 1 (since 0 ≤ γ ≤ 1). □

Now we are ready to provide

Proof

Denote $M={\int \limits }_{B} \rho _{\gamma }$. From Lemmas 2 and 3, we have

$$ \text{Tr}(-{\varDelta}_{B} \gamma) \ge {\int}_{B} |\nabla \sqrt{\rho_{\gamma}}|^{2} \ge \frac{1}{C_{d} M^{2/d}} {\int}_{B} \rho_{\gamma}^{1+2/d} - \frac{C_{d}}{ |B|^{2/d}} M. $$

Combining with (12), we find that for every ε > 0,

$$ (1+\varepsilon) \text{Tr}(-{\varDelta}_{B} \gamma) \ge \frac{\varepsilon}{C_{d} M^{2/d}} {\int}_{B} \rho_{\gamma}^{1+2/d} + \frac{1}{ |B|^{2/d}} \Big[ \frac{M-1}{C_{d}} - \varepsilon C_{d} M\Big]. $$

When M = 2, we can always choose ε = ε_d > 0 such that

$$ \frac{M-1}{C_{d}} - \varepsilon C_{d} M \ge 0, $$

and the desired estimate follows. □

Step 2: In order to put Lemma 1 in good use, we need

Lemma 5

(Besicovitch covering lemma [4, 22]) Let E be a bounded subset of $\mathbb R^{d}$. Let $\mathcal {F}$ be a collection of balls in $\mathbb R^{d}$ such that every point x ∈ E is the center of a ball from $\mathcal {F}$. Then there is a sub-collection $\mathcal {G} \subset \mathcal {F}$ such that

namely E is covered by $\bigcup _{B\in \mathcal G} B$ and every point in E belongs to at most b_d balls from $\mathcal {G}$. Here the constant b_d > 0 depends only on the dimension d ≥ 1 (in particular, b_d is independent of E).

Now we conclude

Proof

By a density argument, it suffices to assume that $\gamma = {\sum }_{i=1}^{N} \lambda _{n} |u_{n}\rangle \langle u_{n}|$ with $N<\infty$, with orthonormal functions $\{u_{n}\}_{n=1}^{N} \subset C_{c}^{\infty }(\mathbb R^{d})$, and with 0 ≤ λ_n ≤ 1.

If ${\int \limits }_{\mathbb R^{d}}\rho _{\gamma }\le 2$, then from Lemma 2 and Sobolev’s inequality (4), we find that

$$ \text{Tr} (-{\varDelta} \gamma) \ge {\int}_{\mathbb R^{d}} |\nabla \sqrt{\rho_{\gamma}}|^{2} \ge \frac{{\int}_{\mathbb R^{d}} \rho_{\gamma}^{1+\frac 2 d}}{ C_{d} ({\int}_{\mathbb R^{d}} \rho_{\gamma})^{2/d}} \ge \frac{1}{C_{d} 2^{\frac 2 d}}{\int}_{\mathbb R^{d}} \rho_{\gamma}^{1+\frac 2 d}. $$

Hence, it remains to consider the case ${\int \limits }_{\mathbb R^{d}} \rho _{\gamma } \ge 2.$ Since ρ_γ is continuous and its support $E\subset \mathbb R^{d}$ is bounded, for every x ∈ E, we can find a ball $B_{x} \subset \mathbb R^{d}$ centered at x such that

$$ {\int}_{B_{x}} \rho_{\gamma} = 2. $$

Applying the Besicovitch covering lemma, from the collection of balls $\mathcal {F}=\{B_{x}\}_{x\in E}$, we can find a sub-collection $\mathcal {G} \subset \mathcal {F}$ such that

The second bound in (13) implies that

$$ b_{d} {\int}_{\mathbb R^{d}} |\nabla u_{n}|^{2} \ge \sum\limits_{B\in \mathcal{G}} {\int}_{B} |\nabla u_{n}|^{2} $$

for all n. Therefore,

$$ \begin{array}{@{}rcl@{}} b_{d} \text{Tr} (-{\varDelta} \gamma) &=& \sum\limits_{n=1}^{N} \lambda_{n} b_{d} {\int}_{\mathbb R^{d}} |\nabla u_{n}|^{2} \ge \sum\limits_{n=1}^{N} \lambda_{n} \sum\limits_{B\in \mathcal{G}} {\int}_{B} |\nabla u_{n}|^{2}\\ &=& \sum\limits_{B\in \mathcal{G}} \sum\limits_{n=1}^{N} \lambda_{n} {\int}_{B} |\nabla u_{n}|^{2} = \sum\limits_{B \in \mathcal G} \text{Tr} (-{\varDelta}_{B} \gamma). \end{array} $$

On the other hand, for every ball $B \subset \mathcal G$, we have ${\int \limits }_{B} \rho _{\gamma } = 2$ and 0 ≤ γ ≤ 1 on L²(B) (since 0 ≤ γ ≤ 1 on $L^{2}(\mathbb R^{d})$). Thus, Lemma 1 gives

$$ \text{Tr}(-{\varDelta}_{B} \gamma) \ge \frac 1 C_{d} {\int}_{B} \rho_{\gamma}^{1+\frac{2}{d}}. $$

(14)

Finally, combining the local bound (13) with the covering property (13), we conclude that

$$ b_{d} \text{Tr} (-{\varDelta} \gamma) \ge \sum\limits_{B \in \mathcal G} \text{Tr} (-{\varDelta}_{B} \gamma) \ge \frac 1 {C_{d}} \sum\limits_{B \in \mathcal G} {\int}_{B} \rho_{\gamma}^{1+2/d} \ge \frac 1 {C_{d}} {\int}_{\mathbb R^{d}} \rho_{\gamma}^{1+2/d}. $$

The proof is complete. □