Determine the sharp constant in the Cwikel-Lieb-Rozenblum bound on the number of bound states of a three-dimensional Schrodinger operator

How many negative-energy bound states can a given three-dimensional potential support? Schwinger's 1961 analysis of this question introduced the integral-kernel criterion now called the Birman-Schwinger principle, proved Bargmann's per-partial-wave bound sharp, and derived the first fully general three-dimensional counting bound under pointwise decay conditions. Later literature transformed the question into the search for the optimal constant in a homogeneous inequality: the number $N(V)$ of negative eigenvalues of $-\Delta+V$ on $L^2(\mathbb R^3)$ satisfies $N(V)\le L_{0,3}\int V(x)_-^{3/2}\,dx$ (the Cwikel-Lieb-Rozenblum bound, the $\gamma=0$ case of the Lieb-Thirring inequality family). The sharp constant is still open after half a century: the best rigorous bounds are $(8/\sqrt3)\,L^{\rm cl}_{0,3}\le L_{0,3}\le 6.86924\,L^{\rm cl}_{0,3}$ with semiclassical constant $L^{\rm cl}_{0,3}=1/(6\pi^2)$, and the standing conjecture is $L_{0,3}=(8/\sqrt3)\,L^{\rm cl}_{0,3}$, the one-particle (Sobolev) value. The problem is to determine $L_{0,3}$ exactly and thereby decide this conjecture.

Background

In quantum mechanics, a single particle in an external potential is described by a Schrodinger operator $H=-\Delta+V$ acting on $L^2(\mathbb R^3)$, in units where $\hbar^2/2m=1$; here $-\Delta$ is the kinetic-energy operator and $V:\mathbb R^3\to\mathbb R$ the potential. The bound states of the system are the negative eigenvalues of $H$; their number is finite for decaying potentials. A basic quantitative question is how large this number can be for a potential of a given strength. Write $V(x)_-=\max(-V(x),0)$ for the negative part of $V$ (only the attractive part can bind), and $N(V)$ for the number of negative eigenvalues of $H$, counted with multiplicity.

The first bounds were angular-momentum-resolved. For a central (spherically symmetric) potential, Bargmann (1952) proved that the number $n_\ell$ of bound states in the partial wave (fixed-angular-momentum channel) of angular momentum $\ell$ satisfies $(2\ell+1)\,n_\ell<\int_0^\infty r\,|V(r)|\,dr$. Schwinger's 1961 paper re-derived this bound by a general variational kernel method, showed it is the best possible (attractive delta-shell potentials, i.e. negative potentials concentrated on an infinitely thin spherical shell, nearly saturate it), extended the method to non-central and spin-tensor potentials (spin-dependent forces coupling the two particles' spins to their relative orbital motion, as in the two-nucleon problem), and obtained the first fully general three-dimensional bound, $N\le(4\pi)^{-2}\iint |V(\mathbf r)|\,|V(\mathbf r')|/|\mathbf r-\mathbf r'|^2\,d\mathbf r\,d\mathbf r'$, valid for potentials decaying faster than $r^{-2}$ at infinity and less singular than $r^{-2}$ at finitely many points. His central tool was an auxiliary compact integral operator, now called the Birman-Schwinger operator, $K_E=|V|^{1/2}(-\Delta-E)^{-1}|V|^{1/2}$ for $E<0$: a negative energy $E$ is an eigenvalue of $H$ if and only if $1$ is an eigenvalue of $K_E$, so counting bound states of $H$ reduces to counting eigenvalues of $K_E$ above $1$. This Birman-Schwinger principle, and the associated coupling-constant argument (scaling $V$ by a parameter $\lambda$ and watching eigenvalues descend through zero), underlies almost all later work on the subject.

The modern form of the question is homogeneous. Semiclassical (Weyl) analysis predicts that for slowly varying potentials the bound-state count is approximately the volume of the classically allowed region of phase space, $N(V)\approx L^{\rm cl}_{0,3}\int V_-^{3/2}$, where the semiclassical constant $L^{\rm cl}_{\gamma,d}=(2\pi)^{-d}\int_{\mathbb R^d}(1-p^2)_+^{\gamma}dp=\Gamma(\gamma+1)/((4\pi)^{d/2}\Gamma(\gamma+d/2+1))$ gives $L^{\rm cl}_{0,3}=1/(6\pi^2)$. The Cwikel-Lieb-Rozenblum (CLR) bound, proved independently around 1976, states that for every $V\in L^{3/2}(\mathbb R^3)$ (no pointwise decay needed), $N(V)\le L_{0,3}\int V_-^{3/2}$ for some finite constant $L_{0,3}$. It holds only in dimensions $d\ge 3$: in $d=1$ and $d=2$, any attractive potential, however weak, binds at least one state, so no bound of this form can hold. Of the three original proofs, the one by Lieb gives the best constant, $L_{0,3}\le 6.86924\,L^{\rm cl}_{0,3}$ (Lieb 1976), and this remains the best upper bound in three dimensions today.

Lieb and Thirring (1975, 1976) embedded the counting bound in a one-parameter family of moment inequalities: $\sum_j|E_j|^{\gamma}\le L_{\gamma,3}\int V(x)_-^{\gamma+3/2}dx$ for $\gamma\ge 0$, where $E_j<0$ are the bound-state energies ($\gamma=0$ recovers the CLR bound). The dual kinetic-energy form of the $\gamma=1$ inequality bounds the kinetic energy of orthonormal families of functions and is what makes the mathematical proof of the stability of matter (the theorem that the ground-state energy of ordinary bulk matter is bounded below in proportion to the number of particles, so that matter does not collapse) possible; it was created for that many-body purpose. The sharp constants were conjectured by Lieb and Thirring to obey $L_{\gamma,d}=\max(L^{\rm cl}_{\gamma,d},L^{\rm one}_{\gamma,d})$, where $L^{\rm one}_{\gamma,d}$ is a 'one-particle' constant produced by a single weakly bound state. In the counting case $\gamma=0$, $d=3$, the maximizer is the one-particle value $L^{\rm one}_{0,3}=(8/\sqrt3)L^{\rm cl}_{0,3}$, which follows from the sharp Sobolev inequality (the optimal constant in the three-dimensional inequality bounding the $L^6$-norm of a function by the $L^2$-norm of its gradient); hence the conjecture $L_{0,3}=(8/\sqrt3)L^{\rm cl}_{0,3}\approx 4.6188\,L^{\rm cl}_{0,3}$, about a factor $1.49$ below the best proven upper bound. The Aizenman-Lieb monotonicity principle states that the ratios $R_{\gamma,d}=L_{\gamma,d}/L^{\rm cl}_{\gamma,d}$ are non-increasing in $\gamma$, so the value of $L_{0,3}$ caps every moment constant in the same dimension.

What is known sharp: the conjecture is proved for $\gamma\ge 3/2$ in all dimensions (there $L_{\gamma,d}=L^{\rm cl}_{\gamma,d}$; Laptev-Weidl 2000) and in the single case $d=1$, $\gamma=1/2$ (Hundertmark-Lieb-Thomas 1998), the only known case where the sharp constant exceeds the semiclassical value. Sharp constant still undetermined: $d=1$ with $1/2<\gamma<3/2$, and every $d\ge 2$ with $0\le\gamma<3/2$, including the counting case $\gamma=0$ in $d=3$ (this problem) and the kinetic-energy case $\gamma=1$ in all dimensions, which controls the exact constants in Thomas-Fermi-type energy bounds for fermionic many-body systems (Thomas-Fermi theory is the mean-field approximation describing large fermionic systems such as heavy atoms). Within this undetermined region the max-conjecture itself is known to fail in parts (Helffer-Robert 1990: $L_{\gamma,d}>L^{\rm cl}_{\gamma,d}$ for $d\ge2$, $0\le\gamma<1$; Glaser-Grosse-Martin 1978: $L_{0,d}>L^{\rm one}_{0,d}$ for $d\ge7$; Frank-Gontier-Lewin 2021: failure for $d=3$ with $1/2<\gamma<\gamma_3^{\rm c}\approx0.8627$), but at $\gamma=0$, $d=3$ the conjecture remains open. The dedicated surveys by Frank (2020) and Schimmer (2022) document precisely this state of the art; Hundertmark, Kunstmann, Ried, and Vugalter (2022) improved the CLR constants in dimensions $d\ge 5$ but not in $d=3$.

Problem Statement

Determine the sharp value of the constant $L_{0,3}$ in the Cwikel-Lieb-Rozenblum inequality: the smallest constant such that for every real-valued $V\in L^{3/2}(\mathbb R^3)$, the number $N(V)$ of negative eigenvalues, counted with multiplicity, of the Schrodinger operator $-\Delta+V$ on $L^2(\mathbb R^3)$ satisfies

$N(V)\;\le\;L_{0,3}\int_{\mathbb R^3}V(x)_-^{3/2}\,dx,$

where $V(x)_-=\max(-V(x),0)$. Currently $(8/\sqrt3)\,L^{\rm cl}_{0,3}\le L_{0,3}\le 6.86924\,L^{\rm cl}_{0,3}$ with $L^{\rm cl}_{0,3}=1/(6\pi^2)$, and the standing conjecture (the $\gamma=0$, $d=3$ instance of the Lieb-Thirring conjecture) is $L_{0,3}=(8/\sqrt3)\,L^{\rm cl}_{0,3}=(8/\sqrt3)/(6\pi^2)$, the one-particle (Sobolev) value.

Determining the sharp value of $L_{0,3}$ requires (i) proving the inequality with the claimed constant for all real $V\in L^{3/2}(\mathbb R^3)$, and (ii) proving optimality by exhibiting potentials for which the ratio $N(V)/\int V_-^{3/2}$ approaches the claimed constant, so that no smaller constant is valid. This settles the conjecture and closes the factor-1.49 gap between the known bounds. The same question - determining the sharp constant - remains open in every dimension $d\ge 3$ at $\gamma=0$ and, for $0\le\gamma<3/2$ and $d\ge 2$, for the moment inequalities (the conjectured value $\max(L^{\rm cl}_{\gamma,d},L^{\rm one}_{\gamma,d})$ is known to fail in parts of that broader region, e.g. at $\gamma=0$ for $d\ge 7$, but the exact constants there are undetermined); resolving the general Lieb-Thirring conjecture would resolve this three-dimensional counting case, whereas partial results (improved bounds without a matching optimality construction, or results in other dimensions or moments only) would not resolve it.

References

1. Julian Schwinger, 'On the Bound States of a Given Potential', Proceedings of the National Academy of Sciences 47(1), 122-129 (1961). DOI: 10.1073/pnas.47.1.122, https://doi.org/10.1073/pnas.47.1.122 (full text open access via EuropePMC, https://europepmc.org/articles/PMC285255).
2. V. Bargmann, 'On the Number of Bound States in a Central Field of Force', Proceedings of the National Academy of Sciences 38(11), 961-966 (1952). DOI: 10.1073/pnas.38.11.961, https://doi.org/10.1073/pnas.38.11.961.
3. Elliott H. Lieb, 'Bounds on the eigenvalues of the Laplace and Schroedinger operators', Bulletin of the American Mathematical Society 82(5), 751-753 (1976). DOI: 10.1090/s0002-9904-1976-14149-3, https://doi.org/10.1090/s0002-9904-1976-14149-3. Announcement of Lieb's CLR proof and best constant.
4. Elliott H. Lieb, 'The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem', Proc. Sympos. Pure Math. 36, 241-252 (1980); reprinted in E. H. Lieb, 'Inequalities: Selecta of Elliott H. Lieb' (Springer, 2002), pp. 243-254. DOI: 10.1007/978-3-642-55925-9_22, https://doi.org/10.1007/978-3-642-55925-9_22. Full paper; best constant $L_{0,3}\le 6.86924\,L^{\rm cl}_{0,3}$.
5. Michael Cwikel, 'Weak Type Estimates for Singular Values and the Number of Bound States of Schrodinger Operators', Annals of Mathematics (2) 106(1), 93-100 (1977). DOI: 10.2307/1971160, https://doi.org/10.2307/1971160. One of the three independent proofs of the CLR bound.
6. Barry Simon, 'Analysis with weak trace ideals and the number of bound states of Schrödinger operators', Transactions of the American Mathematical Society 224(2), 367-380 (1976). DOI: 10.1090/S0002-9947-1976-0423128-X, https://doi.org/10.1090/S0002-9947-1976-0423128-X. Trace-ideal formulation generalizing Schwinger's trace method.
7. G. V. Rozenbljum, 'Distribution of the discrete spectrum of singular differential operators', Soviet Mathematics. Doklady 13, 245-249 (1972) (Russian original: Dokl. Akad. Nauk SSSR 202, 1012-1015 (1972); detailed version: Izv. Vyss. Uchebn. Zaved. Matematika 164, 75-86 (1976), English trans. Sov. Math. (Iz. VUZ) 20, 75-86 (1976)). Zbl: 0249.35069, https://zbmath.org/?q=an%3A0249.35069. The 'R' of the CLR bound.
8. Elliott H. Lieb and Walter E. Thirring, 'Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities', in E. H. Lieb, B. Simon, A. S. Wightman (eds.), Studies in Mathematical Physics (Princeton University Press, 1976), pp. 269-304. DOI (ebook reprint): 10.1515/9781400868940-014, https://doi.org/10.1515/9781400868940-014. The moment inequalities and the sharp-constant conjecture.
9. Elliott H. Lieb and Walter E. Thirring, 'Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter', Physical Review Letters 35(11), 687-689 (1975). DOI: 10.1103/PhysRevLett.35.687, https://doi.org/10.1103/PhysRevLett.35.687. The many-body (dual) kinetic-energy inequality.
10. Michael Aizenman and Elliott H. Lieb, 'On Semi-Classical Bounds for Eigenvalues of Schrodinger Operators', Physics Letters A 66, 427-429 (1978). DOI: 10.1016/0375-9601(78)90385-7, https://doi.org/10.1016/0375-9601(78)90385-7; reprinted in E. H. Lieb, 'The Stability of Matter: From Atoms to Stars' (Springer Selecta, 1991), pp. 170-172, DOI (reprint): 10.1007/978-3-662-02725-7_14, https://doi.org/10.1007/978-3-662-02725-7_14. Monotonicity of $R_{\gamma,d}$ in $\gamma$.
11. V. Glaser, H. Grosse, A. Martin, 'Bounds on the number of eigenvalues of the Schrödinger operator', Communications in Mathematical Physics 59(2), 197-212 (1978). DOI: 10.1007/BF01614249, https://doi.org/10.1007/BF01614249. Physics-line continuation of Schwinger's program; also proves $L_{0,d}>L^{\rm one}_{0,d}$ for $d\ge 7$.
12. Ari Laptev and Timo Weidl, 'Sharp Lieb-Thirring inequalities in high dimensions', Acta Mathematica 184(1), 87-111 (2000). DOI: 10.1007/BF02392782, https://doi.org/10.1007/BF02392782. Sharp constants for $\gamma\ge 3/2$ via the lifting argument.
13. Dirk Hundertmark, Elliott H. Lieb, Lawrence E. Thomas, 'A sharp bound for an eigenvalue moment of the one-dimensional Schroedinger operator', Advances in Theoretical and Mathematical Physics 2, 719-731 (1998). arXiv: math-ph/9806012, https://arxiv.org/abs/math-ph/9806012. The only known sharp constant exceeding the semiclassical value ($d=1$, $\gamma=1/2$).
14. Rupert L. Frank, 'The Lieb-Thirring inequalities: recent results and open problems', arXiv:2007.09326 (2020), https://arxiv.org/abs/2007.09326. Survey; documents best constants, the CLR history, and the open sharp-constant conjectures.
15. Lukas Schimmer, 'The state of the Lieb-Thirring conjecture', arXiv:2203.06051 (2022), https://arxiv.org/abs/2203.06051. Survey; documents the conjecture, known and failing cases, best constants ($(8/\sqrt3)\le R_{0,3}\le 6.86924$), and the Schwinger-Birman attribution.
16. Dirk Hundertmark, Peer Kunstmann, Tobias Ried, Semjon Vugalter, 'Cwikel's bound reloaded', Inventiones mathematicae 231(1), 111-167 (2023). DOI: 10.1007/s00222-022-01144-7, https://doi.org/10.1007/s00222-022-01144-7; arXiv:1809.05069, https://arxiv.org/abs/1809.05069. Improved CLR constants only in $d\ge 5$ (their Table 1: 5.95405 vs 5.96677 in $d=5$); in $d=3$ their bound is 7.55151, so Lieb's 6.86924 remains the best known.
17. Sabine Bögli and Jean-Claude Cuenin, 'Counterexample to the Laptev--Safronov conjecture', Communications in Mathematical Physics 398(3), 1349-1370 (2023). DOI: 10.1007/s00220-022-04546-z, https://doi.org/10.1007/s00220-022-04546-z; arXiv:2109.06135, https://arxiv.org/abs/2109.06135. Refutes the complex-potential analog.
18. 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(3), Article 50 (2025). DOI: 10.1007/s00220-024-05216-y, https://doi.org/10.1007/s00220-024-05216-y; arXiv:2403.04347, https://arxiv.org/abs/2403.04347. Solves the variational problem behind the Hundertmark-Kunstmann-Ried-Vugalter approach; the optimized constants do not improve over Lieb's in $d=3,4$ and improve only for $d\ge 5$, confirming the state of the art in $d=3$.
19. Bernard Helffer and Didier Robert, 'Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture. II', Annales de l'I.H.P. Physique Théorique 53(2), 139-147 (1990). Shows $L_{\gamma,d}>L^{\rm cl}_{\gamma,d}$ for $d\ge 2$, $0\le\gamma<1$.
20. Rupert L. Frank, David Gontier, Mathieu Lewin, 'The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb-Thirring Inequalities', Communications in Mathematical Physics 384, 1783-1828 (2021). DOI: 10.1007/s00220-021-04039-5, https://doi.org/10.1007/s00220-021-04039-5. Disproves the max-conjecture for $d=3$ with $1/2<\gamma<\gamma_3^{\rm c}\approx 0.8627$.