Asymptotics of the ground state energy of heavy atoms and molecules in combined magnetic field

Abstract

We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider $$H=((D-A)\cdot\boldsymbol{\sigma})^2-V$$ with $$V=\sum_{1\le m\le M} \frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$ \mathsf{H}=\sum_{1\le n\le N} H_{x_n} +\sum_{1\le n < n'\le N}|x_n-x_n'|^{-1} $$ on the Fock space $\wedge _{1\le n\le N} L^2(\mathbb{R}^3, \mathbb{C}^2)$. Here $A=A^0+A'$ where $A^0=\frac{1}{2}(-B x_2, B x_1,0)$ is an external magnetic field and $A'$ is self-generated magnetic field. Then the ground state energy is given by $$ \mathsf{E}(A)=\inf \operatorname{Spec}(\mathsf{H})+\frac{1}{\alpha}\int |\nabla \times A'|^2\,dx $$ where the last term is the energy of magnetic field. Under assumptions $\alpha Z\le \kappa^*$ (with a small constant $\kappa^*$) and $M=1$ (atomic case) we study the ground State Energy $$\mathsf{E}^*=\inf_{A'}\mathsf{E}(A).$$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections (depending on $B\ll Z^3$). In the next versions we will consider also molecules and related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.