Sharp critical coupling and sharp stability constants for the pseudo-relativistic many-body Coulomb Hamiltonian

Fefferman and de la Llave (1986) studied the stability of the massless pseudo-relativistic Coulomb Hamiltonian $H=\sum_{j=1}^N\sqrt{-\Delta_{x_j}}+\alpha V$ of $N$ electrons and $M$ fixed nuclei, proving by computer-assisted interval arithmetic that matter with unit nuclear charges is stable for $\alpha<1/(2.06\pi)$, and announcing without proof a small-coupling theorem (their Theorem 2): some $\alpha_{\mathrm{critical}}>0$ exists such that $\alpha<\alpha_{\mathrm{critical}}$ together with $\alpha\max_k Z_k\le 2/\pi$ implies stability; they hoped $\alpha_{\mathrm{critical}}>1/137$ and asked for the best constant $C$ in the derived nonrelativistic stability estimate $(D)$, first for one electron. The deferred Theorem 2 and the $1/137$ question were resolved by Lieb and Yau (1988), who proved stability for $q\alpha\le 1/47$ and $Z\alpha\le 2/\pi$, with subsequent improvements by Lieb, Loss and Siedentop (1996) and Lieb, Siedentop and Solovej (1997); Frank, Lieb and Seiringer (2007) later reached, with arbitrary classical magnetic fields, the full critical nuclear charge $Z\alpha\le 2/\pi$ whenever $q\alpha\le 1/66.5$ — a range covering the physical $\alpha=1/137$ for $q=2$. The surviving open core is the sharp version: (i) the exact critical fine-structure constant $\alpha_c$ — currently confined to $[1/(47q),\,128/(15\pi)]$ without magnetic fields and to $[1/(66.5q),\,128/(15\pi)]$ with magnetic fields, a gap of about two orders of magnitude — and the sharp boundary of the stability region in the $(\alpha,Z_{\max})$ plane, including whether the Lieb–Yau coupling constant $1/(47q)$, or indeed any $\alpha>1/(66.5q)$, can be sustained at the critical nuclear charge once magnetic fields are present; and (ii) the best constant in the nonrelativistic-type estimate $(D)$, beginning with the one-electron case, where molecular configurations already bind more strongly than a single nucleus and no sharp value is known.

Background

Stability of matter is the statement that the ground-state energy of a Coulomb system of $N$ electrons and $M$ nuclei is bounded below by a constant times $N+M$, uniformly in the nuclear positions — the mathematical core of why bulk matter does not collapse. Fefferman and de la Llave (1986) study the pseudo-relativistic (massless) version: the kinetic energy $p^2/2m$ of each electron is replaced by $|p|=\sqrt{-\Delta}$, motivated by the physics of stellar collapse and by Herbst's (1977) spectral analysis of the one-electron relativistic Coulomb operator. The Hamiltonian is $H=\sum_{j=1}^N |p_j|+\alpha V$, where $\alpha>0$ is the fine-structure constant (coupling), $Z_1,\dots,Z_M>0$ are the fixed nuclear charges at positions $y_k\in\mathbb{R}^3$, and $V=\sum_{j<k}|x_j-x_k|^{-1}-\sum_{j,k}Z_k|x_j-y_k|^{-1}+\sum_{i<j}Z_iZ_j|y_i-y_j|^{-1}$ is the full Coulomb potential (electron–electron repulsion, electron–nucleus attraction, nucleus–nucleus repulsion). Because $H$ is dilation-invariant, its lower bound is either $0$ or $-\infty$; 'stability' means $\langle\Psi,H\Psi\rangle\ge 0$ for every antisymmetric normalized $N$-electron wave function $\Psi$ and every nuclear configuration. For a single electron and a single nucleus, Herbst (1977) proved the sharp dichotomy: $|p|-\alpha Z/|x|\ge 0$ if and only if $\alpha Z\le 2/\pi$; with the physical $\alpha=1/137$ this makes $2/(\pi\alpha)\approx 87$ the largest stable point nuclear charge, and complete one-electron collapse occurs for $\alpha Z>2/\pi$. For many bodies the possible behaviors split into stability, many-body implosion (each fixed system bounded below, but the energy tending to $-\infty$ as nuclei merge), and complete implosion; the last occurs exactly when $\alpha\max_k Z_k>2/\pi$. Before 1986, Conlon (1984) had proved many-body stability for the tiny coupling $\alpha\max_k Z_k<10^{-200}$, and Daubechies and Lieb (1983) had settled the one-electron, many-nucleus case under the sharp condition $\alpha\max_k Z_k\le 2/\pi$. Fefferman and de la Llave then proved (Theorem 1, by a computer-assisted interval-arithmetic argument — rigorous machine-verified enclosure of numbers and of solutions of differential equations in intervals — reduced to the non-vanishing of a discriminant attached to a second-order linear ODE for a one-electron, three-nucleus problem) that matter with all $Z_k=1$ is stable for $\alpha<1/(2.06\pi)$, and announced (Theorem 2, proof deferred) that for some $\alpha_{\mathrm{critical}}>0$, stability holds whenever $\alpha<\alpha_{\mathrm{critical}}$ and $\alpha\max_k Z_k\le 2/\pi$; they wrote 'we hope that $\alpha_{\mathrm{critical}}>1/137$' (with numerical evidence due to H. Trotter) and 'In this paper, we prove only Theorem 1, leaving Theorem 2 for later in the hope of getting $\alpha_{\mathrm{critical}}>1/137$.' They also derived from stability, via the elementary operator inequality $|p|\le\varepsilon p^2+\varepsilon^{-1}/4$, the nonrelativistic-type estimate $(D)$,

$$\Big\langle\Psi,\Big(\sum_{j=1}^N(-\Delta_{x_j})+V\Big)\Psi\Big\rangle\ge -C\,N,$$

with $C=\alpha^{-2}/4$ from the trivial argument — a sharp form of the nonrelativistic stability estimate with a linear-in-$N$ constant — and remarked that already for a single electron ($N=1$) the estimate is not obvious (its Thomas–Fermi analogue is false; Thomas–Fermi theory is the semiclassical leading-order approximation to many-body ground-state energies), asking for the best constant in $(D)$ with $N=1$. Later work formulates the many-electron problem with $q$ spin states per electron ($q=2$ in nature; the 1986 paper ignored spin, $q=1$). Lieb and Yau (1988) proved Theorem 2 with explicit constants — stability for $q\alpha\le 1/47$ and $Z\alpha\le 2/\pi$ — together with complementary instability results ($Z_i\alpha\ge 2/\pi$ for some nucleus, or $\alpha>128/(15\pi)\approx 2.72$ with many nuclei, or bosonic statistics). Lieb, Loss and Siedentop (1996) gave a proof via relativistic Thomas–Fermi comparison, valid with arbitrary classical magnetic fields, showing stability if $(\pi/2)Z+2.2159\,q^{1/3}Z^{2/3}+1.0307\,q^{1/3}\le 1/\alpha$; for $q=2$ this permits $\alpha$ up to $0.77$ as $Z\to 0$ and $Z\le 59$ at the physical $\alpha=1/137$, versus the one-electron-optimal $87$; they leave reaching $137\cdot 2/\pi$ with magnetic fields as an explicit challenge. Frank, Lieb and Seiringer (2007) met that challenge at small coupling: they extended the Lieb–Yau argument to arbitrary classical magnetic fields, proving stability for all nuclear charges up to the critical value $\alpha Z_j\le 2/\pi$ whenever $q\alpha\le 1/66.5$ (for $q=2$ this covers $\alpha$ up to $1/133>1/137$, so at the physical fine-structure constant the largest rigorously stable nuclear charge with magnetic fields is again $87=\lfloor 137\cdot 2/\pi\rfloor$); the price is a coupling constant weaker than Lieb–Yau's $1/(47q)$. Lieb, Siedentop and Solovej (1997) settled the stability of relativistic matter interacting with classical magnetic fields in the Brown–Ravenhall formulation (electrons restricted to the positive-energy spectral subspace of a Dirac operator), showing instability for every coupling when the positive-energy projection uses the free Dirac operator. The modern reference treatment of this circle of problems is the chapter 'Stability of Relativistic Matter' in Lieb and Seiringer's monograph The Stability of Matter in Quantum Mechanics.

Problem Statement

Determine the sharp stability constants of the pseudo-relativistic many-body Coulomb Hamiltonian

$$H_{N,M}=\sum_{j=1}^{N}\sqrt{-\Delta_{x_j}}+\alpha V,\qquad V=\sum_{j<k}\frac{1}{|x_j-x_k|}-\sum_{j,k}\frac{Z_k}{|x_j-y_k|}+\sum_{i<j}\frac{Z_iZ_j}{|y_i-y_j|},$$

with $N$ electrons having $q$ spin states, $M$ fixed nuclei of charges $Z_k\in(0,Z_{\max}]$ at arbitrary positions $y_k\in\mathbb{R}^3$, and coupling $\alpha>0$; 'stable' means $\langle\Psi,H_{N,M}\Psi\rangle\ge 0$ for every antisymmetric normalized $\Psi$ and every nuclear configuration. Two coupled sharp questions remain open.

(1) Critical coupling and stability region: determine $\alpha_c=\sup\{\alpha: H_{N,M}\text{ is stable for all }N,M\text{ and all charges with }\alpha Z_{\max}\le 2/\pi\}$ and, more generally, the exact boundary of the stability region in the $(\alpha,Z_{\max})$ plane, both without and with classical magnetic fields (i.e. with each kinetic energy $\sqrt{-\Delta_{x_j}}$ replaced by $\sqrt{(-i\nabla_{x_j}+A(x_j))^2}$ for an arbitrary vector potential $A$). Without magnetic fields the current rigorous bounds are $1/(47q)\le\alpha_c\le 128/(15\pi)\approx 2.72$ (Lieb–Yau). With magnetic fields, Frank–Lieb–Seiringer (2007) achieve the full critical nuclear charge $Z_{\max}=2/(\pi\alpha)$ — at the physical $\alpha=1/137$ this is the one-electron-optimal $87=\lfloor 137\cdot 2/\pi\rfloor$ for $q=2$ — but only for $q\alpha\le 1/66.5$, a weaker coupling constant than Lieb–Yau's $1/(47q)$, and nothing is known with fields for $\alpha>1/(66.5q)$. The target is the exact value of $\alpha_c$ and the exact boundary of the stability region in either setting.

(2) Best constant in the nonrelativistic-type estimate $(D)$: under stability of the relativistic Hamiltonian, the elementary inequality $\sqrt{-\Delta}\le\varepsilon(-\Delta)+\varepsilon^{-1}/4$ yields

$$\Big\langle\Psi,\Big(\sum_{j=1}^{N}(-\Delta_{x_j})+V\Big)\Psi\Big\rangle\ge -C\,N$$

with $C=\alpha^{-2}/4$ from the trivial argument; determine the best (smallest) constant $C^{*}$ for the field-free model — a function of the maximal nuclear charge $Z_{\max}$ (under stability $Z_{\max}\le 2/(\pi\alpha)$, so for the relativistic model with coupling $\alpha$ the relevant value is $C^{*}$ at $Z_{\max}=2/(\pi\alpha)$, to be compared with the trivial $\alpha^{-2}/4$) — in the one-electron case

$$C^{*}_{N=1}=\sup_{M\ge 1}\ \sup_{0<Z_k\le Z_{\max}}\ \sup_{y_1,\dots,y_M}\Big[-\inf\sigma\Big(-\Delta_x-\sum_{k}\frac{Z_k}{|x-y_k|}\Big)-\sum_{i<j}\frac{Z_iZ_j}{|y_i-y_j|}\Big],$$

i.e. the worst-case binding of one nonrelativistic electron by arbitrarily many nuclei of charge at most $Z_{\max}$, net of their mutual Coulomb repulsion. Fefferman and de la Llave note that even for $N=1$ this is not obvious and its Thomas–Fermi analogue is false; already two unit nuclei (the $H_2^+$ molecular ion) bind one electron more strongly than a single nucleus, so the sharp constant exceeds the hydrogenic value $Z_{\max}^2/4$ and the extremal configuration is unknown.

References

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