The asymptotics for the number of eigenvalue branches for the magnetic Schrödinger operator \(H-\lambda W\) in a gap of \(H\) (Q1358304)

Let \(H\) be a Schrödinger operator with a uniform magnetic and bounded electric potential such that \(\sigma(H)\) has gaps, and let \(E\) be in a gap. Let \(W\geq 0\) be a potential decaying at infinity. In this paper, the asymptotics as \(\lambda\to+\infty\), of the numbers \(N_{\pm}(\lambda; H-E;W)\) of the eigenvalue branches of families \(H\mp\lambda W\) crossing an energy level \(E\) is computed. For \(+\) sign, the case of \(W\) decaying as \(|x|^{-2}\) or slowly is considered, for \(-\) sign, any polynomial decay is admissible.