Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields

Abstract

We consider the Schrödinger operator with magnetic field, H = ( 1 i ∇ − a ⇀ ( x ) ) 2 + V ( x )  in  R n . \begin{equation*}H=(\frac {1}{i}\nabla -{\overset {\rightharpoonup }{a}}(x))^{2}+V(x) \text { in } \mathbb {R}^{n}. \end{equation*} Assuming that V ≥ 0 V\ge 0 and | curl a ⇀ | + V + 1 |\text {curl}\, \overset {\rightharpoonup }{a}|+V+1 is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.