Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields (Q1266223)

The authors investigate the Pauli operator perturbed by an electric field falling off at infinity. In general the unperturbed Pauli operator, acting in \(L^2(\mathbb{R}^3)\otimes\mathbb{C}^2\), has the form \[ H_p= (-i\nabla- A)^2- \sigma\cdot B \] with magnetic potential \(A: \mathbb{R}^3\to \mathbb{R}^3\), the vector \(\sigma= (\sigma_1,\sigma_2,\sigma_3)\) of \(2\times 2\) Pauli matrices and the magnetic field \(B= \nabla\times A\). Here it is assumed that \(B= (0,0,b)\) has constant direction, where the function \(b= b(x)\) is positive and decays at infinity. This implies that \(H_p\) can be reduced to an operator \(H\) acting on \(L^2(\mathbb{R}^2)\) and it is known that \(H\) has no essential spectrum in the interval \((-\infty,0)\). The perturbed Pauli operator is given by \(H(V)= H-V\) with electric field \(V= V(x)\) decaying at infinity faster than \(b\). Under such assumptions, the authors obtain formulas in terms of \(b\) and \(V\) for the asymptotic distribution of negative eigenvalues of the operator \(H(V)\).