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Lieb-Thirring inequalities and other functional inequalities for orthonormal systems · 2022-ana-200

Functional Analysis · ICM 2022: Lieb-Thirring inequalities and other functional inequalities for orthonormal systems

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Problem statement

For each integer d≥1d\geq 1, let KdK_d be the largest constant such that, for every integer N≥1N\geq 1 and every family u1,…,uN∈H1(Rd)u_1,\ldots,u_N\in H^1(\mathbb R^d) that is orthonormal in L2(Rd)L^2(\mathbb R^d),

∑n=1N∫Rd∣∇un(x)∣2 dx≥Kd∫Rd(∑n=1N∣un(x)∣2)1+2/d dx. \sum_{n=1}^N\int_{\mathbb R^d}|\nabla u_n(x)|^2\,dx \geq K_d\int_{\mathbb R^d}\left(\sum_{n=1}^N|u_n(x)|^2\right)^{1+2/d}\,dx.

Let Kd(1)K_d^{(1)} be the largest constant such that

∫Rd∣∇u(x)∣2 dx≥Kd(1)∫Rd∣u(x)∣2+4/d dx(∫Rd∣u(x)∣2 dx)−2/d \int_{\mathbb R^d}|\nabla u(x)|^2\,dx \geq K_d^{(1)}\int_{\mathbb R^d}|u(x)|^{2+4/d}\,dx \left(\int_{\mathbb R^d}|u(x)|^2\,dx\right)^{-2/d}

for every nonzero u∈H1(Rd)u\in H^1(\mathbb R^d), and define

Kdcl=dd+2(2π)2ωd−2/d,ωd=∣{x∈Rd:∣x∣<1}∣. K_d^{\mathrm{cl}}=\frac{d}{d+2}(2\pi)^2\omega_d^{-2/d}, \qquad \omega_d=\bigl|\{x\in\mathbb R^d:|x|<1\}\bigr|.

Determine whether

Kd=min⁡{Kd(1),Kdcl} K_d=\min\{K_d^{(1)},K_d^{\mathrm{cl}}\}

holds for every integer d≥1d\geq 1.

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Question 1

Let K1K_1 be the largest constant such that, for every integer N≥1N\geq1 and every family u1,…,uN∈H1(R)u_1,\ldots,u_N\in H^1(\mathbb R) that is orthonormal in L2(R)L^2(\mathbb R),

∑n=1N∫R∣un′(x)∣2 dx≥K1∫R(∑n=1N∣un(x)∣2)3 dx. \sum_{n=1}^N\int_{\mathbb R}|u_n'(x)|^2\,dx \geq K_1\int_{\mathbb R}\left(\sum_{n=1}^N|u_n(x)|^2\right)^3\,dx.

Let K1(1)K_1^{(1)} be the largest constant such that

∫R∣u′(x)∣2 dx≥K1(1)∫R∣u(x)∣6 dx(∫R∣u(x)∣2 dx)−2 \int_{\mathbb R}|u'(x)|^2\,dx \geq K_1^{(1)}\int_{\mathbb R}|u(x)|^6\,dx \left(\int_{\mathbb R}|u(x)|^2\,dx\right)^{-2}

for every nonzero u∈H1(R)u\in H^1(\mathbb R). Prove or disprove that K1=K1(1)K_1=K_1^{(1)}.

Source reference: Lukas Schimmer, “The state of the Lieb--Thirring conjecture,” in The Physics and Mathematics of Elliott Lieb: The 90th Anniversary, Vol. II, EMS Press, 2022, pp. 253--275, Section 4.1 (“Lieb--Thirring conjecture: d=1d=1” and “dual version for γ=1\gamma=1”), DOI: https://doi.org/10.4171/90-2/39.

Question 2

Let K2K_2 be the largest constant such that, for every integer N≥1N\geq1 and every family u1,…,uN∈H1(R2)u_1,\ldots,u_N\in H^1(\mathbb R^2) that is orthonormal in L2(R2)L^2(\mathbb R^2),

∑n=1N∫R2∣∇un(x)∣2 dx≥K2∫R2(∑n=1N∣un(x)∣2)2 dx. \sum_{n=1}^N\int_{\mathbb R^2}|\nabla u_n(x)|^2\,dx \geq K_2\int_{\mathbb R^2}\left(\sum_{n=1}^N|u_n(x)|^2\right)^2\,dx.

Let K2(1)K_2^{(1)} be the largest constant such that

∫R2∣∇u(x)∣2 dx≥K2(1)∫R2∣u(x)∣4 dx(∫R2∣u(x)∣2 dx)−1 \int_{\mathbb R^2}|\nabla u(x)|^2\,dx \geq K_2^{(1)}\int_{\mathbb R^2}|u(x)|^4\,dx \left(\int_{\mathbb R^2}|u(x)|^2\,dx\right)^{-1}

for every nonzero u∈H1(R2)u\in H^1(\mathbb R^2). Prove or disprove that K2=K2(1)K_2=K_2^{(1)}.

Source reference: Lukas Schimmer, “The state of the Lieb--Thirring conjecture,” in The Physics and Mathematics of Elliott Lieb: The 90th Anniversary, Vol. II, EMS Press, 2022, pp. 253--275, Section 4.1 (“Lieb--Thirring conjecture: dual version for γ=1\gamma=1”), DOI: https://doi.org/10.4171/90-2/39.

Question 3

Let K3K_3 be the largest constant such that, for every integer N≥1N\geq1 and every family u1,…,uN∈H1(R3)u_1,\ldots,u_N\in H^1(\mathbb R^3) that is orthonormal in L2(R3)L^2(\mathbb R^3),

∑n=1N∫R3∣∇un(x)∣2 dx≥K3∫R3(∑n=1N∣un(x)∣2)5/3 dx. \sum_{n=1}^N\int_{\mathbb R^3}|\nabla u_n(x)|^2\,dx \geq K_3\int_{\mathbb R^3}\left(\sum_{n=1}^N|u_n(x)|^2\right)^{5/3}\,dx.

Define

K3cl=35(2π)2ω3−2/3,ω3=∣{x∈R3:∣x∣<1}∣. K_3^{\mathrm{cl}}=\frac35(2\pi)^2\omega_3^{-2/3}, \qquad \omega_3=\bigl|\{x\in\mathbb R^3:|x|<1\}\bigr|.

Prove or disprove that K3=K3clK_3=K_3^{\mathrm{cl}}.

Source reference: Lukas Schimmer, “The state of the Lieb--Thirring conjecture,” in The Physics and Mathematics of Elliott Lieb: The 90th Anniversary, Vol. II, EMS Press, 2022, pp. 253--275, Section 4.1 (“Lieb--Thirring conjecture: dual version for γ=1\gamma=1”), DOI: https://doi.org/10.4171/90-2/39.

Question 4

Prove or disprove that every pair of real-valued functions u1,u2∈H1(R)u_1,u_2\in H^1(\mathbb R) satisfying

∫Rui(x)uj(x) dx=δij(i,j∈{1,2}) \int_{\mathbb R}u_i(x)u_j(x)\,dx=\delta_{ij} \qquad (i,j\in\{1,2\})

obeys

∫R(∣u1′(x)∣2+∣u2′(x)∣2) dx≥π24∫R(u1(x)2+u2(x)2)3 dx. \int_{\mathbb R}\bigl(|u_1'(x)|^2+|u_2'(x)|^2\bigr)\,dx \geq \frac{\pi^2}{4}\int_{\mathbb R}\bigl(u_1(x)^2+u_2(x)^2\bigr)^3\,dx.

Source reference: Rafael D. Benguria and Michael Loss, “Connection between the Lieb--Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane,” in Partial Differential Equations and Inverse Problems, Contemporary Mathematics 362, American Mathematical Society, 2004, pp. 53--61, Equation (3.4), DOI: https://doi.org/10.1090/conm/362/06604; arXiv:math-ph/0402048.

Question 5

Let Γ\Gamma be a smooth closed plane curve of length 2π2\pi, parametrized by arc length s∈R/(2πZ)s\in\mathbb R/(2\pi\mathbb Z), and assume that its curvature κ(s)\kappa(s) is strictly positive. On L2(R/(2πZ))L^2(\mathbb R/(2\pi\mathbb Z)), consider the periodic Schrödinger operator

HΓ=−d2ds2+κ(s)2. H_\Gamma=-\frac{d^2}{ds^2}+\kappa(s)^2.

Prove or disprove that the lowest eigenvalue of HΓH_\Gamma is at least 11.

Source reference: Rafael D. Benguria and Michael Loss, “Connection between the Lieb--Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane,” in Partial Differential Equations and Inverse Problems, Contemporary Mathematics 362, American Mathematical Society, 2004, pp. 53--61, Section 3 (Ovals Conjecture), DOI: https://doi.org/10.1090/conm/362/06604; arXiv:math-ph/0402048.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

Verdict: unresolved.

The problem is the dual kinetic-energy form of the Lieb--Thirring sharp-constant conjecture at eigenvalue exponent γ=1\gamma=1. If L1,dL_{1,d} denotes the optimal constant in

Tr⁡(−Δ+V)−≤L1,d∫RdV−1+d/2, \operatorname{Tr}(-\Delta+V)_{-}\leq L_{1,d}\int_{\mathbb R^d}V_-^{1+d/2},

duality gives Kd/Kdcl=(L1,d/L1,dcl)−2/dK_d/K_d^{\mathrm{cl}}=(L_{1,d}/L_{1,d}^{\mathrm{cl}})^{-2/d}, with analogous relations for the one-particle constants. Thus the stated equality is exactly L1,d=max⁡{L1,d(1),L1,dcl}L_{1,d}=\max\{L_{1,d}^{(1)},L_{1,d}^{\mathrm{cl}}\}. Current sources still identify this γ=1\gamma=1 conjecture as open in every dimension. More specifically, the minimum in the statement is Kd(1)K_d^{(1)} for d=1,2d=1,2 and KdclK_d^{\mathrm{cl}} for d≥3d\geq3, so the outstanding equalities are Kd=Kd(1)K_d=K_d^{(1)} in dimensions 1,21,2 and Kd=KdclK_d=K_d^{\mathrm{cl}} in every dimension d≥3d\geq3.

Key references and established progress:

  • Lukas Schimmer, “The state of the Lieb--Thirring conjecture,” in The Physics and Mathematics of Elliott Lieb: The 90th Anniversary, Vol. II, EMS Press, 2022, pp. 253--275, DOI 10.4171/90-2/39. Section 2 gives the exact dual relation between KdK_d and L1,dL_{1,d}, and Section 4.1 states the dual γ=1\gamma=1 conjecture in the same form as this problem and records that it is open in all dimensions. It also explains the above split between d≤2d\leq2 and d≥3d\geq3.

  • Rupert L. Frank, David Gontier, and Mathieu Lewin, “The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb--Thirring inequalities,” Communications in Mathematical Physics 384 (2021), 1783--1828, DOI 10.1007/s00220-021-04039-5, arXiv:2002.04964. Their Theorem 1 implies, at γ=1\gamma=1 and d≥3d\geq3, that every finite-rank Lieb--Thirring constant is strictly smaller than the unrestricted eigenvalue constant. Their Theorem 2 also shows that the two-bound-state constant exceeds the one-bound-state constant in these dimensions. These results describe the many-particle character of any optimizer but do not prove that the unrestricted constant is semiclassical. The strict hypotheses in dimensions 11 and 22 exclude the present endpoint γ=1\gamma=1.

  • Thiago Carvalho Corso and Tobias Ried, “On a Variational Problem Related to the Cwikel--Lieb--Rozenblum and Lieb--Thirring Inequalities,” Communications in Mathematical Physics 406 (2025), article 50, DOI 10.1007/s00220-024-05216-y, arXiv:2403.04347. Corollary 1.7 optimizes the Frank--Hundertmark--Jex--Nam method and proves L1,1/L1,1cl≤1.44655L_{1,1}/L_{1,1}^{\mathrm{cl}}\leq1.44655; dimension lifting gives the same ratio bound for every d≥1d\geq1. Equivalently, Kd/Kdcl≥1.44655−2/dK_d/K_d^{\mathrm{cl}}\geq1.44655^{-2/d}. This is a quantitative improvement, not the conjectured equality.

  • Rupert L. Frank, Dirk Hundertmark, Michal Jex, and Phan Thành Nam, “The Lieb--Thirring inequality revisited,” Journal of the European Mathematical Society 23 (2021), no. 8, 2583--2600, DOI 10.4171/JEMS/1062, arXiv:1808.09017. This paper introduced the direct kinetic-energy estimate subsequently optimized by Carvalho Corso and Ried and substantially improved the known lower bounds on KdK_d, but did not determine the sharp constant.

  • Ari Laptev and Timo Weidl, “Sharp Lieb--Thirring inequalities in high dimensions,” Acta Mathematica 184 (2000), 87--111, DOI 10.1007/BF02392782. They prove the semiclassical sharp constant for eigenvalue exponents γ≥3/2\gamma\geq3/2 in all dimensions by dimension lifting. This major exact result lies outside the present exponent γ=1\gamma=1 and therefore does not close the gap.

  • Giao Ky Duong, Thi Minh Thao Le, Phan Thành Nam, and Phuoc-Tai Nguyen, “Finite-Rank Optimizers for the Mass--Supercritical Lieb--Thirring and Hardy--Lieb--Thirring Inequalities,” preprint, 2025, arXiv:2510.24148. Its introduction explicitly records, after the 2025 literature, that the mass-critical sharp-constant conjecture considered here remains open in all dimensions; the paper’s optimizer results concern the distinct mass-supercritical regime.

Consequently, no cited theorem proves the required equality for all dd, and no counterexample at the exact exponent γ=1\gamma=1 is known. Results disproving parts of the broader Lieb--Thirring conjecture at other exponents do not resolve this statement.

References
  1. [1]ICM Conjectures · 2022-ana-200 link
  2. [2]ICM 2022: Lieb-Thirring inequalities and other functional inequalities for orthonormal systems link

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