Asymptotics of the ground state energy of heavy molecules in self-generated 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 $$H=((D-A)\cdot \mathbf{\sigma }{)}^2-V$$ with $$V=\sum _{1\le m\le M}\frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$\mathsf{H}=\sum _{1\le n\le N}{H}_{x_n}+\sum _{1\le n<n^{\prime }\le N}|x_n-x_{n^{\prime }}{|}^{-1}$$ on the Fock space ${\wedge }_{1\le n\le N}{L}^2({\mathbb{R}}^3,{\mathbb{C}}^2)$. Here $A$ is a self-generated magnetic fiels. Then the ground state energy is given by $$\mathsf{E}(A)=\inf \mathrm{Spec}(\mathsf{H})+\frac{1}{\alpha }\int |\nabla \times A{|}^{2}dx$$ where the last term is the energy of magnetic field. Under assumption $\alpha Z\le {\kappa }^{\ast }$ (with a small constant ${\kappa }^{\ast }$) we study the ground State Energy $${\mathsf{E}}^{\ast }=\underset{A}{\inf }\mathsf{E}(A).$$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections. We also consider related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.

81 pp. This preprint supersedes my preprints arXiv:1108.4188, arXiv:1112.2487 and arXiv:1112.5538 as it contains more detailed analysis, more results, stronger results and also some corrections. v3: Corrected an error in v1, 2; v.4: Corrected misprint in abstract

References in corpus (3)

• The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction
• Ground state energy of large atoms in a self-generated magnetic field
• Sharp Spectral Asymptotics for Operators with Irregular Coefficients. III. Schroedinger operator with a strong magnetic field