The Magnetisation of Large Atoms¶in Strong Magnetic Fields

In this paper we study the asymptotic form of the magnetisation and current of large atoms in strong constant magnetic fields. We prove that the Magnetic Thomas–Fermi theory gives the right magnetisation/current for magnetic field strengths which satisfy B≤Z ^4/3. Received: 24 April 2000 / Accepted: 21 August 2000     

On the Ginzburg‐Landau critical field in three dimensions

We study the three‐dimensional Ginzburg‐Landau model of superconductivity. Several “natural” definitions of the (third) critical field, H, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian with constant magnetic field and with Neumann (magnetic) boundary condition in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis, we give an affirmative answer to a conjecture by Pan. © 2008 Wiley Periodicals, Inc.     

On the Semiclassical Asymptotics of the Current and Magnetic Moment of a Non-Interacting Electron Gas at Zero Temperature in a Strong Constant Magnetic Field

We calculate the asymptotic form of the quantum current and magnetic moment of a non-interacting electron gas at zero temperature. The calculation uses coherent states and a novel commutator identity for the current operator.     

Existence of surface smectic states of liquid crystals

The Landau–de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state.     

Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field

We consider a large neutral molecule with total nuclear charge Z in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z α ² ≤ κ ₀ for a sufficiently small κ ₀, where α denotes the fine structure constant. We show that, in the simultaneous limit Z → ∞, α → 0 such that κ = Z α ² is fixed, the ground state energy of the system is given by a two term expansion c ₁ Z ^7/3 + c ₂(κ) Z ² + o(Z ²). The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion.     

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy bòB²{\beta \int B^{2}} and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h ² ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h^1+e{h^{1+\varepsilon}} , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.     

Nucleation of bulk superconductivity close to critical magnetic field

We consider the two-dimensional Ginzburg–Landau functional with constant applied magnetic field. For applied magnetic fields close to the second critical field HC₂ and large Ginzburg–Landau parameter, we provide leading order estimates on the energy of minimizing configurations. We obtain a fine threshold value of the applied magnetic field for which bulk superconductivity contributes to the leading order of the energy. Furthermore, the energy of the bulk is related to that of the Abrikosov problem in a periodic lattice. A key ingredient of the proof is a novel L^∞-bound which is of independent interest.     

Analyticity of the density of electronic wavefunctions

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic inR ³ away from the nuclei.     

Spectral confinement and current for atoms in strong magnetic fields

We study confinement of the ground state of atoms in strong magnetic fields to different subspaces related to the lowest Landau band. Using the results on confinement we can calculate the quantum current in the entire semiclassical region B?Z³.