[Paper Review] Real analyticity away from the nucleus of pseudorelativistic Hartree-Fock orbitals

Anna Dall’Acqua, Søren Fournais+2 more|arXiv (Cornell University)|Mar 25, 2011

Spectral Theory in Mathematical Physics11 references3 citations

TL;DR

This paper establishes the real analyticity of pseudorelativistic Hartree-Fock orbitals away from the nucleus, using a modified version of the Morrey-Nirenberg nested balls technique adapted to handle the non-local pseudodifferential operator, singular potential, and non-linear terms. The key result is that the quantum mechanical ground state of pseudorelativistic atoms cannot be a Hartree-Fock state, extending a non-relativistic result to the relativistic setting.

ABSTRACT

We prove that the Hartree--Fock orbitals of pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator sqrt{-Delta+1}-1, are real analytic away from the origin. As a consequence, the quantum mechanical ground state of such atoms is never a Hartree-Fock state. Our proof is inspired by the classical proof of analyticity by nested balls of Morrey and Nirenberg. However, the technique has to be adapted to take care of the non-local pseudodifferential operator, the singularity of the potential at the origin, and the non-linear terms in the equation.

Motivation & Objective

• To establish the real analyticity of Hartree-Fock orbitals in pseudorelativistic atoms, where electron kinetic energy is modeled by the operator √(−Δ+1)−1.
• To extend the non-relativistic result—where analyticity implies the ground state is not a Hartree-Fock state—to the pseudorelativistic case.
• To adapt the classical Morrey-Nirenberg nested balls method to handle the non-local nature of the pseudodifferential operator and the singular Coulomb potential.
• To prove that smooth solutions to the non-linear pseudorelativistic Hartree-Fock equation are real analytic away from the origin, under general conditions on the potential and non-linearity.

Proposed method

• Adaptation of the Morrey-Nirenberg nested balls technique to the non-local pseudodifferential setting, incorporating singular potentials and non-linear terms.
• Use of weighted Sobolev spaces and interpolation inequalities (e.g., Gagliardo-Nirenberg) to control higher-order norms and bootstrap regularity.
• Application of elliptic regularity theory for variable-coefficient operators to derive higher integrability and Hölder continuity.
• Employment of a priori estimates in $L^{3p}$ and $W^{2,3p/2}$ spaces to control non-linear terms and propagate analyticity.
• Use of embedding theorems (e.g., Morrey’s inequality) to upgrade $L^p$-regularity to Hölder continuity and ultimately to real analyticity.
• Construction of a sequence of balls with shrinking radii to propagate analyticity from a given point away from the nucleus, leveraging the structure of the Euler-Lagrange equation.

Experimental results

Research questions

• RQ1 Are Hartree-Fock orbitals in pseudorelativistic atoms real analytic away from the nucleus, despite the non-local kinetic energy operator and singular Coulomb potential?
• RQ2 Can the classical Morrey-Nirenberg analyticity method be extended to non-local, non-linear equations with singularities and non-smooth coefficients?
• RQ3 Does the analyticity of orbitals in the pseudorelativistic case imply that the true quantum ground state cannot be a Hartree-Fock state, as in the non-relativistic case?
• RQ4 What conditions on the potential and non-linearity allow the same analyticity result to be generalized beyond the specific pseudorelativistic model?
• RQ5 How does the presence of a point singularity at the origin affect the propagation of analyticity in solutions to non-linear equations?

Key findings

• The Hartree-Fock orbitals for pseudorelativistic atoms are real analytic in $\mathbb{R}^3 \setminus \{0\}$, despite the non-local kinetic energy operator $\sqrt{-\Delta + 1} - 1$ and the singular $1/|\mathbf{x}|$ potential.
• The quantum mechanical ground state of a pseudorelativistic atom is never a Hartree-Fock state, as the orbitals are analytic away from the nucleus and thus cannot be finite linear combinations of Slater determinants.
• The proof technique extends to a broader class of equations of the form $(-\Delta + m)^s \varphi + V\varphi + |\varphi|^k \varphi = \lambda \varphi$, provided $V$ has finitely many point singularities and satisfies certain regularity and decay conditions.
• Smooth solutions to the non-linear equation $\big(\sqrt{-\Delta + 1}\big)\varphi - \frac{Z}{|\cdot|}\varphi \pm \big(|\varphi|^2 * |\cdot|^{-1}\big)\varphi = \lambda\varphi$ are real analytic away from the origin.
• The method relies on iterative $L^p$-estimates, interpolation inequalities, and a modified nested ball argument to propagate analyticity, even in the presence of non-local and singular terms.
• The result holds under the physical constraint $Z\alpha \leq 2/\pi$, ensuring the Hamiltonian is bounded from below.