Sharp one-dimensional Lieb–Thirring constants

1 Introduction

The Lieb–Thirring inequality bounds the negative eigenvalues of a Schrödinger operator by an integral of its potential. Write t_+=\max\{t,0\} and t_-=\max\{-t,0\}. On the line, let 0\le W\in L^{\gamma+1/2}(\R) and let H_{-W} be the self-adjoint operator associated with the quadratic form

For \gamma>1/2, this form is closed and bounded below, and its negative spectrum is discrete. We give the needed form and compactness arguments in Section 7. Write

counting multiplicities; before the inequality is proved this sum is allowed to be infinite. The optimal constant is

Two natural comparisons are the semiclassical constant

and the optimal constant when only the lowest eigenvalue is retained, denoted by L_{\gamma,1}^{(1)}. The one-dimensional Lieb–Thirring conjecture predicts the one-bound-state constant for 1/2<\gamma<3/2 and the semiclassical constant for \gamma\ge3/2.

The estimate includes rough, unbounded potentials and infinitely many negative eigenvalues. Corollary 7.3 gives the corresponding inequality for signed potentials on their natural dense weighted form domain. The theorem concerns scalar potentials on the line; the matrices used below encode finite orthonormal families. Its sharpness assertion exhibits an equality potential without classifying all optimizers.

1.1 History and relation to prior work

Lieb and Thirring introduced these spectral inequalities in their work on the kinetic energy of fermions and the stability of matter [15, 16]. At exponent one, optimizing over the potential makes the spectral inequality equivalent to a lower bound on the kinetic energy of an orthonormal family in terms of its density. On the line, for a finite orthonormal family (u_j)\subset H^1(\R), this density is \rho=\sum_j|u_j|^2, and the corresponding kinetic bound controls \int\rho^3 by \sum_j\int|u_j'|^2. The one-state variational problem goes back to Keller [12]. The conjecture asks whether allowing arbitrarily many bound states can improve on its constant in the lower exponent range. We verify the one-state equality potential and its normalization directly.

The scalar upper endpoint predates the general Lieb–Thirring inequalities. Inverse-scattering trace identities for the Korteweg–de Vries equation give the sharp value L_{3/2,1}=3/16 [22, 8]; Lieb and Thirring explicitly credit Gardner, Greene, Kruskal and Miura in [16, Section 4(B)]. Aizenman and Lieb’s monotonicity argument extends the semiclassical equality to all larger exponents [1]. At the other endpoint, Weidl first proved finiteness of L_{1/2,1} [21], and Hundertmark, Lieb and Thomas obtained its sharp value 1/2 [11, Theorem 1]. Laptev and Weidl established the operator-valued upper-endpoint estimate and used dimension lifting to obtain the sharp semiclassical constants for every dimension and \gamma\ge3/2 [13, Theorems 2.1, 2.2 and 3.1].

Quantitative estimates and orthonormal-function methods narrowed the remaining gap. Hundertmark, Laptev and Weidl proved L_{\gamma,1}\le2L_{\gamma,1}^{\cl} for 1/2\le\gamma<3/2 [10, Theorem 4.1]. At exponent one, Dolbeault, Laptev and Loss extended the method of Eden and Foias to matrix-valued potentials [6, 5]. Their all-dimensional semiclassical ratio was \pi/\sqrt3; subsequent bounds were 1.456 by Frank, Hundertmark, Jex and Nam [7, Theorem 1] and the reported 1.44655 of Corso and Ried [3, Corollary 1.7]. These bounds preceded the sharp exponent-one theorem described below. In the full one-dimensional strict interval, Levitt’s numerical investigation observed convergence to the one-state profile or splitting into increasingly separated copies [14, Section 4.1]. Such computations support the conjectured value but do not bound the contribution of every orthonormal family. That uniform bound is the obstacle addressed by our action inequality.

Read and Schulz proved the case \gamma=1 in [19, Theorem 1], obtaining the sharp constant 4/(3\sqrt3\pi). Their proof extends Benguria and Loss’s scalar cumulative-mass argument [2, Section 2] to orthonormal families through a matrix correction to the kinetic energy. Their operator-valued extension also gives the semiclassical ratio 2/\sqrt3 in every dimension at exponent one [19, Theorem 2 and Section 2]. Our proof retains the individual eigenvalue scales in a finite-interval action inequality. Its main analytic step is a rank-one comparison for an energy-dependent regularized matrix field. A continuation argument chooses a lower triangular coefficient matrix and a path joining two prescribed matrix endpoints. Together these yield the action inequality for every 1/2<\gamma<3/2. Theorem 1.1 therefore resolves the remaining open-interval cases of the one-dimensional Lieb–Thirring conjecture positively; the case \gamma=1 is already covered by Read–Schulz.

1.2 Proof strategy

Put \sigma=\gamma-1/2\in(0,1). The central result is a variational inequality on a bounded interval, independent of any potential. Fix I=(x_-,x_+). Given N orthonormal real functions u_i\in H^1_0(I;\R) and positive numbers k_i, let u=(u_i)_{i=1}^N and K=\diag(k_i). We prove that the action

has supremum at least \sum_i\int_{-k_i}^{k_i}(k_i^2-s^2)^\sigma\,ds among functions a=Cu with C lower triangular. Section 2 states this precisely in Theorem 2.1 and explains the role of triangularity.

The difficulty is to obtain this lower bound for an arbitrary orthonormal family while retaining its individual scales k_i. The argument replaces a direct optimization over C by a path with prescribed matrix endpoints. Section 3 constructs a regularized map B\mapsto M_\delta(B) from real symmetric N\times N matrices to themselves, and a scalar primitive whose change from K to -K gives the required endpoint integral. Section 4 proves the analytic comparison that controls the nonlinear action term when M_\delta(B)=aa^{\transpose}. Its integrated tangent map is an average of unitary conjugations; a trace inequality removes this average from the comparison.

Section 5 chooses C and a path from K to -K by continuation modulo row signs. Compactness and the exclusion of zero rows make the continuation possible even when C loses rank. Section 6 combines the path with the primitive: failure of the rank-one constraint contributes a negative square to the action identity. Compactness allows the penalty parameter to tend to zero at fixed regularization. Only then is the regularization removed.

Section 7 returns to the potential. Take the u_i to be Dirichlet eigenfunctions with eigenvalues -k_i^2, ordered so that k_1\ge\cdots\ge k_N. Lower triangularity lets the eigenfunction equations bound the quadratic part of the action by \int W|a|^2. A scalar maximization bounds the whole action by a multiple of \int W^{1+\sigma}. Finally, cutoff trial spaces pass from bounded intervals to the line, and finite partial sums exhaust all negative eigenvalues. The sharpness calculation uses the explicit potential in Theorem 1.1.

2 The finite-interval action inequality

Fix

All matrices below are finite dimensional. Write \Sym_N for the real symmetric N\times N matrices, with inner product \tr(AB) and norm \|A\|_{\HS}=(\tr(A^2))^{1/2}. The notation A\le B means that B-A is positive semidefinite. Vectors are columns, and |a| is their Euclidean norm.

Let I=(x_-,x_+) be a bounded interval. For a positive diagonal matrix K=\diag(k_1,\ldots,k_N) and a\in H^1_0(I;\R^N), define

In one dimension H^1_0(I) embeds into the continuous functions, so every term in this integral is finite.

The restriction to lower triangular matrices is what makes this inequality useful spectrally. If the u_i are eigenfunctions ordered from the lowest eigenvalue upward, the ith component of Cu stays in the span of the first i eigenfunctions. Section 7 will use this observation to bound the left side from above by an integral of the potential.

Here is the construction behind the lower bound. For each \delta>0, Section 3 builds a locally Lipschitz map M:\Sym_N\to\Sym_N and a C^1 function J:\Sym_N\to\R such that

They have two decisive properties. First, Section 4 proves

Second, Section 5 finds, for every \varepsilon>0, a lower triangular C and a path B satisfying

The small parameter \varepsilon penalizes failure of the rank-one constraint. For a=Cu, differentiating J(B)+a^{\transpose}Ba along this path produces the negative term -\varepsilon^{-1}\|M(B)-aa^{\transpose}\|_{\HS}^2. Uniform bounds on the path and on C then give

Finally, the boundary difference tends to the right side of (2.2) as \delta\downarrow0. We keep \delta fixed until after the limit \varepsilon\downarrow0.

3 A regularized matrix field

We construct the field and primitive used in Section 2. Throughout this section K=\diag(k_1,\ldots,k_N)>0 and \delta>0 are fixed. The scalar case explains the choice of field. For N=1, write K=(k). If -k\le b\le k, putting m=(k^2-b^2)^\sigma gives

Thus this field gives equality in the scalar version of the rank-one comparison, and a primitive with derivative -m has the required change \int_{-k}^k(k^2-b^2)^\sigma\,db from k to -k. We seek matrix versions of the comparison and endpoint identity, with a regularization removed at the end.

The identity

suggests integrating a scalar function of the quadratic on the left: its affine dependence on b will let us construct a trace primitive even when the corresponding matrices K and B do not commute. Set \beta=\sigma-\tfrac12\in(-\tfrac12,\tfrac12) and define f:\R\to\R by

The exponent and normalization are chosen so that, for z\ge0, scaling the integration variable gives

Integration in z therefore produces the regularized power (z+\delta)^\sigma-\delta^\sigma. On the scalar interval [-k,k], we will recover this expression with z=k^2-b^2. Keeping f' constant for negative arguments extends the construction beyond that interval, as needed for the continuation argument. The resulting f is C^1, strictly increasing, and concave on \R. The integral defining c is finite because \beta<1/2; the definition also includes \beta=0 without a limiting convention.

For B\in\Sym_N, put

Here and below scalar functions of symmetric or Hermitian matrices are defined by the spectral theorem. The symmetric cutoff in (3.2) is part of the definition.

3.1 Convergence and a primitive

We first record an elementary estimate that keeps the regularity argument in the Hilbert–Schmidt norm.

3.2 Scalar comparison and boundary values

Define

This integral is absolutely convergent: on bounded z-sets its integrand has tail O(|s|^{2\beta-2}).

In particular M(K)=M(-K)=0. We shall also use the following symmetry. If S is a diagonal matrix with entries \pm1, then SKS=K and

This follows already at each cutoff. The matrix K stays fixed throughout; we do not require equivariance under conjugations that move K.

4 The rank-one comparison

The field M was chosen so that its rank-one values control the nonlinear term in the action. We now prove that property. The parameters K,\sigma and \delta remain fixed as in Section 3.

The proof compares X_s(B) with a matrix that is a function of B alone. Its tangent map, integrated in s, is an average of unitary conjugations. Trace monotonicity then removes this average from the needed inequality.

4.1 A tangent inequality on positive matrices

Divided differences describe the derivative of matrix functional calculus, as in the work of Daleckii and Krein [4]; see [9, Theorem 2.3.1] for a modern finite-dimensional formulation. We prove the required derivative formula and tangent inequality directly from the resolvent representation below.

4.2 The integrated tangent map

Suppose that M(B)=aa^{\transpose} and m=|a|^2>0. By Lemma 3.4, for every unit e,

Strict increase of \mu implies e^{\transpose}K^2e\ge(e^{\transpose}Be)^2. Consequently

for every real s: the matrices X_s(B) are positive semidefinite.

To apply the tangent inequality, we compare X_s(B) with (sI-B)^2+dI, which is diagonal in an eigenbasis of B. Choose d>0 so that its integrated scalar value is \mu(d)=m:

This choice matches the nonzero eigenvalue of M(B). When we pair the integrated tangent inequality with M(B), the identity M(B)^2=mM(B) will cancel the contribution mI against the left side. We have X_s(B)=Y_s+(D-dI); we do not assert that D\ge0.

Apply Lemma 4.2 to X_s(B)\ge0 and Y_s>0, and denote the tangent map at Y_s by T_s. Subtract f(s^2)I and integrate symmetrically. In an eigenbasis of B, the shift identity (3.8) gives c\int_\R(f(Y_s)-f(s^2)I)\,ds=mI, with the integral understood at symmetric cutoffs. Thus

The integral of the tangent maps converges absolutely: for large |s|, their matrix coefficients are O(|s|^{2\beta-2}).

The integrated tangent inequality (4.7) controls \mathcal P(D), while the desired conclusion involves D itself. The next step compares them in precisely the trace pairing with M(B).

4.3 Trace monotonicity and contraction

5 A matrix path with prescribed endpoints

The action argument needs a symmetric matrix path from K to -K. Its differential equation also contains a lower triangular coefficient matrix C, which we are free to choose. We choose C by matching the terminal endpoint. At the start of a deformation, the matching problem has exactly one solution after identifying each row of C with its negative. A compactness argument shows that this solution count cannot disappear modulo two.

Fix \delta>0 throughout the remaining construction. We use the matrix field M=M_\delta and scalar function \mu=\mu_\delta constructed above. Recall the properties needed here. The field M is locally Lipschitz on \Sym_N, the space of real symmetric N\times N matrices. The function \mu is locally Lipschitz and strictly increasing, with \mu(0)=0. For K=\operatorname{diag}(k_1,\ldots,k_N) and every diagonal sign matrix S,

For every real unit vector e,

Finally, writing e_i for the ith standard basis vector,

The equivariance in (5.1) only concerns sign matrices, which commute with the fixed matrix K.

The parity principle is the compact-one-manifold argument underlying mod-two degree; see [17, Section 4]. The smooth regular-value facts used below are given in [17, Sections 2–3]. We prove the needed relative version for continuous families of bundle sections here. A section of a vector bundle assigns to each base point a vector in the fiber above it. A zero is transverse if the derivative of the section in any local trivialization maps onto that fiber.

For the regularized field M_\delta, the bound (5.10) may depend on \delta. With \delta fixed, it implies that all pairs (B(x),Cu(x)) produced for 0<\varepsilon\leq1 lie in one compact subset of \Sym_N\times\mathbb R^N. This is the compactness needed in the action limit; it requires no bound on B' uniform in \varepsilon.

6 From the matrix path to the action

We now prove Theorem 2.1. The matrix comparison controls points where M(B)=aa^{\transpose} exactly. The differential equation supplies a penalty for the remaining points, and compactness makes that penalty effective without requiring convergence of the chosen coefficients C.

7 From the action inequality to spectral moments

We apply Theorem 2.1 first to ordered Dirichlet eigenfunctions. We then pass to the line using finite-dimensional trial spaces. This last step treats all negative eigenvalues, without assuming in advance that their moment is finite.

Throughout this section, 0<\sigma<1, p=1+\sigma, q=1+1/\sigma, and \gamma=\sigma+1/2. Write

where \mathrm B denotes the beta function.

7.1 Quadratic forms and negative spectrum

For 0\leq W\in L^p(\R), define

We record the facts needed to define its operator and to exhaust its negative eigenvalues. We use the association of a densely defined closed semibounded form with its self-adjoint operator, and the spectral description of the operator and its form domain, in [20, Theorem 2.13, Theorems 3.7–3.8 and (3.50)–(3.51)]. All theorem numbers here refer to the 2009 first edition.

7.2 The Dirichlet estimate

The order of the eigenfunctions is what makes the triangular coefficient restriction in Theorem 2.1 useful.

The beta–gamma identity identifies the constant without any limiting argument in \sigma:

7.3 Exhaustion of the line

Let 0\leq W\in L^p(\R). If there are no negative eigenvalues, the trace inequality is immediate. Otherwise fix any finite number N of them, counted with multiplicity and ordered increasingly:

Let \phi_1,\ldots,\phi_N\in H^1(\R) be corresponding orthonormal eigenfunctions. Choose smooth cutoffs \chi_R equal to one on [-R,R], supported in I_R=(-2R,2R), and satisfying 0\leq\chi_R\leq1 and \|\chi_R'\|_\infty\leq c/R. Then

Hölder’s inequality and the embedding H^1\subset L^{2p'} show that the potential form is continuous under this convergence.

For clarity, let G_R and Q_R be the Gram and form matrices of the N functions v_{j,R}. Then

in matrix norm. In particular the functions are linearly independent for large R. If g_R=\|G_R-I_N\|_{\mathrm{op}}<1 and e_R=\|Q_R-\Lambda\|_{\mathrm{op}}, their Rayleigh quotients differ uniformly from the limiting coefficient-space quotients by at most

Applying the form-domain min–max principle [20, Theorem 4.10 and the following form-domain extension] to the span of v_{1,R},\ldots,v_{j,R} gives, for every j\leq N,

where \lambda_j(I_R) is the jth Dirichlet eigenvalue on I_R. All these levels are negative for sufficiently large R. Proposition 7.2 now implies

Letting R\to\infty proves the estimate for every finite partial sum. Taking their supremum gives

Thus the full moment is finite even when there are infinitely many negative eigenvalues. No approximation of W by smoother potentials is needed.

7.4 Signed potentials

To include unbounded positive parts, we specify the form domain in the signed-potential statement.

7.5 Sharpness

Put r=1/\sigma>1 and consider

Direct differentiation gives

The function \psi and its first two derivatives decay exponentially. Thus \psi\in H^2(\R), the operator domain for this bounded W, and H_{-W}\psi=-\psi. Substitution t=rx, followed by z=\tanh t, gives

The eigenvalue -1 gives \Tr(H_{-W})_-^\gamma\geq1; the upper bound (7.5) gives the reverse inequality. This completes the proof of Theorem 1.1. Hence this potential attains equality, and the constant in () is optimal. This establishes L_{\gamma,1}=L_{\gamma,1}^{(1)} throughout 1/2<\gamma<3/2.