arXiv subjects
Publications and source records attributed to Heinz Siedentop.
For heavy atoms (large atomic number $Z$) described by no-pair operators in the Furry picture we find the ground state's leading energy correction. We compare the result with (semi-)empirical values and Schwinger's prediction showing more than qualitative agreement.
We show that in graphene charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power.
We investigate a semiclassical momentum density energy functional for atoms and show that it yields the same value as the well-known Thomas-Fermi functional. In fact, we show an explicit relation between the minimizers of the two functionals. The main result is that the atomic momentum density converges on the scale $Z^{2/3}$ to the minimizer of the momentum density energy functional. This allows to determine the linear response of atoms on momentum dependent perturbations.
We show that $$J_{a,b,n}:=\frac12(|p|^a|q|^b+|q|^b|p|^a)$$ is positive, if $n\geq b+a$. (Here $q$ is the multiplication by $x$ and $p:= \mathrm{i}^{-1}\nabla$.) Furthermore we show that it generalizes the generalized Hardy inequalities for the fractional Laplacians.
We show that for the straightforward quantized relativistic Coulomb Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum dot -- the maximal number of electrons does not exceed twice the nuclear charge. It result is then generalized to the presence of external magnetic fields and atomic Hamiltonians. This is based on the positivity of $$|\bx| T(\bp) + T(\bp) |\bx| $$ which -- in two dimensions -- is false for the non-relativistic case $T(\bp) = \bp^2$, but is proven in this paper for $T(\bp) = |\bp|$, i.e., the ultra-relativistic kinetic energy.
Given a 2-dimensional no-pair Weyl operator with a point nucleus of charge Z, we show that a homogeneous magnetic field does not lower the critical charge beyond which it collapses.
We study $N$ interacting massless Dirac fermions confined in a two-dimensional quantum dot. Physical realizations of this problem include a graphene monolayer and the surface state of a strong topological insulator. We consider both a magnetic confinement and an infinite mass confinement. The ground state energy is computed as a function of the effective interaction parameter $α$ from the Hartree-Fock approximation and, alternatively, by employing the Müller exchange functional. For N=2, we compare those approximations to exact diagonalization results. The Hartree-Fock energies are highly accurate for the most relevant interaction range $α\alt 2$, but the Müller functional leads to an unphysical instability when $α\agt 0.756$. Up to 20 particles were studied using Hartree-Fock calculations. Wigner molecule formation was observed for strong but realistic interactions, accompanied by a rich peak structure in the addition energy spectrum.
The increasing interest in the Mueller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix γ(X, X') is replaced by the square of γ^{1/2}(X, X'). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing γ's have unique densities ρ(x), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \leq Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of γ, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.
The ground state energy of the Mueller functional of large (neutral) atoms of atomic number Z agrees with the quantum mechanical ground state energy up to order o(Z to the 5/3). ----- Die Grundzustandsenergie des Muellerfunktionals von grossen (neutralen) Atomen der Ordnungszahl Z stimmt mit der quantenmechanischen Grundzustandsenergie bis zur Ordnung o(Z hoch 5/3) ueberein.
We consider relativistic many-particle operators which - according to Brown and Ravenhall - describe the electronic states of heavy atoms. Their ground state energy is investigated in the limit of large nuclear charge and velocity of light. We show that the leading quasi-classical behavior given by the Thomas-Fermi theory is raised by a subleading correction, the Scott correction. Our result is valid for the maximal range of coupling constants, including the critical one. As a technical tool, a Sobolev-Gagliardo-Nirenberg-type inequality is established for the critical atomic Brown-Ravenhall operator. Moreover, we prove sharp upper and lower bound on the eigenvalues of the hydrogenic Brown-Ravenhall operator up to and including the critical coupling constant.
We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability.
We describe atoms by a pseudo-relativistic model that has its origin in the work of Chandrasekhar. We prove that the leading energy correction for heavy atoms, the Scott correction, exists. It turns out to be lower than in the non-relativistic description of atoms. Our proof is valid up to and including the critical coupling constant. It is based on a renormalization of the energy whose zero level we adjust to be the ground-state energy of the corresponding non-relativistic problem. This allows us to roll the proof back -- by relatively simple technical means -- to results for the Schrödinger operator.
It is shown that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $α^2$Ry.
Starting from a formal Hamiltonian as found in the physics literature -- omitting photons -- we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the one-electron subspace is well-defined. Our construction is non-perturbative and does not use a cut-off. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms.
We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics.
The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields is already unstable when the fine structure constant, is too large it is noteworthy that the combination of the two is still stable provided the projection onto the positive energy states of the Dirac operator, which defines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here.
Stability of matter with Coulomb forces has been proved for non-relativistic dynamics, including arbitrarily large magnetic fields, and for relativistic dynamics without magnetic fields. In both cases stability requires that the fine structure constant alpha be not too large. It was unclear what would happen for both relativistic dynamics and magnetic fields, or even how to formulate the problem clearly. We show that the use of the Dirac operator allows both effects, provided the filled negative energy `sea' is defined properly. The use of the free Dirac operator to define the negative levels leads to catastrophe for any alpha, but the use of the Dirac operator with magnetic field leads to stability.
The strong Scott conjecture about the electron density at a distance 1/Z from an atomic nucleus of charge $Z$ and its generalization for molecules are proved. The density, suitably scaled, converges to an explicit limiting density as $Z \to \infty$. Both a weak convergence and a pointwise convergence on spheres holds.