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

arXiv1312.7533MaRDI QIDQ6247689

Victor Ivrii

Publication date: 29 December 2013

Abstract: We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider ${\displaystyle }$H=((D-A)cdot�oldsymbol{sigma})^2-V${\displaystyle }$ with ${\displaystyle }$V=sum_{1le mle M} frac{Z_m}{|x-y_m|}${\displaystyle }$ and a corresponding Multiparticle Quantum Hamiltonian ${\displaystyle }$ mathsf{H}=sum_{1le nle N} H_{x_n} +sum_{1le n < n'le N}|x_n-x_n'|^{-1} ${\displaystyle }$ on the Fock space $wedg{e}_{1lenleN}{L}^2(mathbb{R}^3,mathbb{C}^2)$. Here $A=A^0+A^{\prime }$ where $A^0=frac12(-B{x}_2,B{x}_1,0)$ is an external magnetic field and $A^{\prime }$ is self-generated magnetic field. Then the ground state energy is given by ${\displaystyle }$ mathsf{E}(A)=inf operatorname{Spec}(mathsf{H})+frac{1}{alpha}int | abla imes A'|^2,dx ${\displaystyle }$ where the last term is the energy of magnetic field. Under assumptions $alphaZlekapp{a}^{*}$ (with a small constant $kapp{a}^{*}$) and $M=1$ (atomic case) we study the ground State Energy ${\displaystyle }$mathsf{E}^*=inf_{A'}mathsf{E}(A).${\displaystyle }$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections (depending on $Bll{Z}^3$). 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.

Mathematics Subject Classification ID

Asymptotic distributions of eigenvalues in context of PDEs (35P20) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20)

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