Sharp upper bounds for the density of relativistic atoms: Noninteracting case

Abstract: We prove an optimal upper bound for the density of electrons of an infinite Bohr atom (no electron-electron interactions) described by the relativistic operators of Chandrasekhar and Dirac. We also consider densities in each angular momentum channel separately.

• The paper establishes order-sharp upper bounds for electron densities in relativistic atomic models using the Chandrasekhar and Dirac-Coulomb operators.
• It utilizes heat kernel analysis and Hardy-type inequalities along with angular momentum decomposition to capture both the singular behavior near the nucleus and the semiclassical decay at large distances.
• The results underpin a rigorous derivation of the strong Scott correction for heavy atoms, paving the way for advances in models of interacting systems and higher-dimensional quantum mechanics.

Sharp Upper Bounds for Electron Densities in Relativistic Atomic Models

Overview of Main Results

The paper establishes optimal, order-sharp upper bounds for the spatial electron density in atomic systems described by the Chandrasekhar and Dirac-Coulomb operators, focusing on the noninteracting (infinite Bohr atom) case. The density bounds are shown to be best possible both near the nucleus and at large distances, and are proven for all angular momentum channels individually. The results address longstanding open questions in the context of the strong Scott correction for relativistic atoms, providing improved estimates that are critical for the asymptotic analysis of large atomic systems.

Chandrasekhar-Coulomb Operator: Density Bounds

Let $C_\kappa = \sqrt{1-\Delta} - 1 - \kappa |x|^{-1}$ denote the Chandrasekhar operator for a hydrogenic atom with effective nuclear charge $\kappa$ in three dimensions. The density of electrons is given by $1_{(-\infty,0)}(C_\kappa)(x,x)$, which sums the squares of all eigenfunctions with negative eigenvalues.

Theorem: For $0 < \kappa \leq 2/\pi$, there exists $A_\kappa < \infty$ such that

$$1_{(-\infty,0)}(C_\kappa)(x,x) \leq A_\kappa \left( |x|^{-2\eta_\kappa} \, 1(|x|\leq 1) + |x|^{-\frac32} 1(|x|>1) \right),$$

where $\eta_\kappa$ is uniquely determined by $$(1-\eta_\kappa)\tan\frac{\pi\eta_\kappa}{2} = \kappa$$.

This bound is shown to be sharp for $|x| \leq 1$, matching the lower bound given by the ground state eigenfunction's behavior. For $|x| > 1$, the bound matches the semiclassical decay known from nonrelativistic models, specifically the $\kappa$0 decay proven by Heilmann and Lieb [HeilmannLieb1995].

The proof utilizes heat kernel analysis for fractional Schrödinger operators with Hardy-type potentials, extending to a general class of operators in arbitrary dimension and order. The method avoids angular momentum decomposition for the main results but also delivers sharp bounds in each angular momentum channel, with explicit dependence of the singularity on the angular momentum quantum number.

Dirac-Coulomb Operator: Density Bounds

For the Dirac-Coulomb operator $\kappa$1 (with $\kappa$2),

$\kappa$3

the density is $\kappa$4, summing the squares of eigenfunctions with eigenvalues in $\kappa$5, relevant for the strong Scott correction in the Furry picture.

Theorem: For $\kappa$6, there exists $\kappa$7 such that

$\kappa$8

where $\kappa$9.

The bound for small $1_{(-\infty,0)}(C_\kappa)(x,x)$0 reflects the singularity of the relativistic density near the nucleus, dictated by the ground state behavior $1_{(-\infty,0)}(C_\kappa)(x,x)$1, and is demonstrated as best possible. At large distances, the density again decays as $1_{(-\infty,0)}(C_\kappa)(x,x)$2, matching the nonrelativistic limit.

The proofs leverage explicit representations of Dirac eigenfunctions (in terms of Laguerre polynomials and confluent hypergeometric functions) and detailed asymptotics in each angular momentum channel. The decomposition into partial waves allows for sharp estimates summable over the angular quantum numbers.

Angular Momentum Channel Analysis

The paper provides sharp bounds on electron densities for fixed angular momentum channels. For Chandrasekhar and fractional Schrödinger operators, leveraging sharp Hardy–Kato–Herbst inequalities, the singularity exponent at the origin decreases with increasing angular momentum, and the effective "critical" values for the coupling constant become less restrictive.

Quantitative bounds in angular channels allow for summability and uniform control necessary for global density estimates. In particular, for large angular momenta, detailed estimates demonstrate a rapid decay in density, enabling uniform summation over the full range of channels.

Implications for Scott Correction and Atomic Theory

The results provide crucial technical ingredients for the rigorous derivation of the strong Scott correction in relativistic atomic models, extending Lieb’s conjecture to the relativistic setting. The improved pointwise bounds are robust with respect to the parameters, allowing tight control of error terms in asymptotic expansions, and directly affect the analysis of many-electron atoms in the large $1_{(-\infty,0)}(C_\kappa)(x,x)$3 (atomic number) limit.

Importantly, the order-sharp exponents found in the bounds are critical for determining the existence, finiteness, and positivity of scaling limits in electron density, facilitating future work on subtle features of density convergence and subleading energy corrections.

From a theoretical perspective, the methods open the way for further exploration of Hardy-type potentials in nonlocal operators, fractional quantum mechanics, and their spectral properties—areas with numerous open questions. Practically, precise density bounds serve as a foundation for improved numerical treatments and effective model building in relativistic quantum chemistry and condensed matter.

Prospects and Future Directions

• Extension to interacting systems: While the present results apply to noninteracting models, techniques developed may pave the way for bounds in interacting cases, where electron-electron repulsion is present.
• Refinement of semiclassical limits: The methods facilitate analysis of semiclassical limits for relativistic operators, potentially yielding corrections relevant for heavy-atom theory.
• Fractional and higher-dimensional analogues: The generalization to operators of arbitrary order and dimension invites application to models with fractional kinetic terms and higher-dimensional quantum systems.
• Stronger convergence and existence results: The bounds serve as a basis for proving strong convergence results and existence of scaling limits for densities—further strengthening the mathematical foundation of atomic structure theory.

Conclusion

The paper establishes optimal, fully explicit upper bounds for electron densities in relativistic atomic models governed by Chandrasekhar and Dirac operators. The results solve substantial technical challenges in asymptotic analysis for heavy atoms, clarify the nature of relativistic singularities in densities, and set the stage for future advances in both mathematical atomic theory and its applications in quantum physics.