On electrons and nuclei in a magnetic field (Q1354665)

The author proves the following result: Main Theorem. Let \(\Gamma\), \(Z>0\) be given. For constants \(c(\Gamma,Z)\), \(C_1(\Gamma, Z)\), \(C_2(\Gamma, Z)\) depending only on \(\Gamma\) and \(Z\), the following holds. Let \(A(x)\) be a vector potential, and let \(L>0\) be a length. Assume \(L< c(\Gamma, Z)\). Define \[ B(x)= \text{curl }A(x)\text{\quad(the magnetic field)}, \] \[ \overline B_L(x)= \Biggl({4\pi\over 3} L^3\Biggr)^{-1} \int_{| y-x|< L} B(y)dy\text{ \quad(the mean of \(B\) over a ball about \(x\))}, \] \[ \begin{multlined} S_L(x)= \Biggl\{\Biggl({4\pi\over 3}L^3\Biggr)^{- 1}\int_{| y-x|< L}| B(y)-\overline B_L(x)|^2dy\Biggr\}^{1/2}\\ \text{(the standard deviation of \(B\) on a ball about \(x\))}.\end{multlined} \] Let \(y_1,\dots, y_M\in\mathbb{R}^3\) and \(Z_1,\dots, Z_M\geq 0\) be given. Assume \(Z_k\leq Z\) for all \(k= 1,2,\dots, M\). Define \[ D(x)= \min\{| x-y_k|: 1\leq k\leq M\}\quad\text{for }x\in \mathbb{R}^3, \] and \[ \begin{multlined} V_{\text{Coulomb}}(x_1\cdots x_N)=\\ \sum_{1\leq j<k\leq N}| x_j- x_k|^{-1}+ \sum_{1\leq j<k\leq M} Z_jZ_k| y_j- y_k|^{-1}- \sum^N_{j= 1} \sum^M_{k=1} Z_k| x_j- y_k|^{-1}.\end{multlined} \] Finally, let \(\psi\in L^2(\mathbb{R}^{3N},(\mathbb{C}^2)^{\otimes N})\) be antisymmetric with norm \(1\), and let \[ H_{\text{Pauli}}= \sum^N_{k= 1} [\sigma^{(k)} (i\nabla_{x_k}- A(x_k))]^2+ V_{\text{Coulomb}}\quad\text{(Pauli Hamiltonian)}. \] Then \[ \langle H_{\text{Pauli}}\psi, \psi\rangle+ \Gamma \int_{\mathbb{R}^3}| B(x)|^2 e^{-D(x)/L}dx+ C_1(\Gamma, Z) \int_{\mathbb{R}^3} (S_L(x))^2 e^{-D(x)/L}dx\geq - C_2(\Gamma,Z){M\over L}. \] {}.