Real analyticity of solutions to Schrödinger equations involving fractional Laplacians
We prove analyticity of solutions in $\mathbb{R}^{n}$, $n\ge1$, to certain nonlocal linear Schrödinger equations with analytic potentials.
math.AP↗
arXiv subjects
Edgardo Stockmeyer
Publications and source records attributed to Edgardo Stockmeyer.
Hydrogen-like atoms in relativistic QED
In this review we consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. For both models we discuss the semi-boundedness of the Hamiltonian, the strict positivity of the ionization energy, and the exponential localization in position space of spectral subspaces corresponding to energies below the ionization threshold. Moreover, we prove the existence of degenerate ground state eigenvalues at the bottom of the spectrum of the Hamiltonian in both models. All these results hold true, for arbitrary values of the fine-structure constant and the ultra-violet cut-off, and for a general class of electrostatic potentials including the Coulomb potential with nuclear charges less than (sometimes including) the critical charges without radiation field. Apart from a detailed discussion of diamagnetic inequalities in QED (which are applied to study the semi-boundedness) all results stem from earlier articles written by the authors. While a few proofs are merely sketched, we streamline earlier proofs or present alternative arguments at many places.
math-ph↗
Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot
We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V=0. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized.
math-ph↗
Real analyticity away from the nucleus of pseudorelativistic Hartree-Fock orbitals
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.
math-ph↗
Existence of ground states of hydrogen-like atoms in relativistic QED II: The no-pair operator
We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultra-violet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model. For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones.
math-ph↗
Existence of ground states of hydrogen-like atoms in relativistic QED I: The semi-relativistic Pauli-Fierz operator
We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of the semi-relativistic Pauli-Fierz operator and prove that the infimum of the spectrum of the latter operator is an eigenvalue. In particular, we verify that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true for arbitrary values of the fine-structure constant and the ultra-violet cut-off as long as the Coulomb coupling constant (i.e. the product of the fine-structure constant and the nuclear charge) is less than 2/π.
math-ph↗
On the non-relativistic limit of a model in quantum electrodynamics
We consider a (semi-)relativistic spin-1/2 particle interacting with quantized radiation. The Hamiltonian has the form $\hat{H}_c^V:=\{c^2[(\mathbf{p}+{\bf A})^2+{\bf σ}\cdot{\bf B}]+(mc^2)^2\}^{1/2}-mc^2+V+H_f$. Assuming that the potential $|V|$ is bounded with respect to the momentum $|\mathbf{p}|$, we show that $\hat{H}_c^V$ converges in norm-resolvent sense to the usual Pauli-Fierz operator when $c$, the speed of light, tends to $\infty$.
math-ph↗
Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics
We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, $e^2$, and the ultra-violet cut-off, $Λ$, and for all nuclear charges less than the critical charge without radiation field, $Z_c=e^{-2}2/(2/π+π/2)$. We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of $e^2$ and $Λ$ and for nuclear charges less than $e^{-2}2/π$.
math-ph↗
On the eigenfunctions of no-pair operators in classical magnetic fields
We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1,\infty) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ<1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a<(1-λ^2)^{1/2}, for λ\in[0,1), and a<1, for λ<0.
math-ph↗
Spectral theory of no-pair Hamiltonians
We prove a HVZ theorem for a general class of no-pair Hamiltonians describing an atom or positively charged ion with several electrons in the presence of a classical external magnetic field. Moreover, we show that there exist infinitely many eigenvalues below the essential spectrum and that the corresponding eigenfunctions are exponentially localized. The novelty is that the electrostatic and magnetic vector potentials as well as a non-local exchange potential are included in the projection determining the model. As a main technical tool we derive various commutator estimates involving spectral projections of Dirac operators with external fields. Our results apply to all nuclear charges less than or equal to 137.
math-ph↗
Hartree-Fock theory for pseudorelativistic atoms
We study the Hartree-Fock model for pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator \sqrt{(pc)^2+(mc^2)^2}-mc^2. We prove the existence of a Hartree-Fock minimizer, and prove regularity away from the nucleus and pointwise exponential decay of the corresponding orbitals.
math-ph↗
On the convergence of eigenfunctions to threshold energy states
We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of H = T-lambda*V as the energy goes to an energy threshold. We do this for three choices of kinetic energy T, namely the non-relativistic Schr"odinger operator, the pseudorelativistc operator sqrt{-Δ+m^2}-m, and the Dirac operator.
math-ph↗
An operator representation for Matsubara sums
In the context of the imaginary-time formalism for a scalar thermal field theory, it is shown that the result of performing the sums over Matsubara frequencies associated with loop Feynman diagrams can be written, for some classes of diagrams, in terms of the action of a simple linear operator on the corresponding energy integrals of the Euclidean theory at T=0. In its simplest form the referred operator depends only on the number of internal propagators of the graph. More precisely, it is shown explicitly that this \emph{thermal operator representation} holds for two generic classes of diagrams, namely, the two-vertex diagram with an arbitrary number of internal propagators, and the one-loop diagram with an arbitrary number of vertices. The validity of the thermal operator representation for diagrams of more complicated topologies remains an open problem. Its correctness is shown to be equivalent to the correctness of some diagrammatic rules proposed a few years ago.
hep-ph↗