Bounds and Gaps of Positive Eigenvalues of Magnetic Schr\"{o}dinger Operators with No or Robin Boundary Conditions

Abstract

We consider magnetic Schr\"{o}dinger operators on a bounded region $\Omega$ with the smooth boundary $\partial \Omega$ in Euclidean space ${\mathbb R}^d$. In reference to the result from Weyl's asymptotic law and P\'{o}lya's conjecture, P. Li and S. -T. Yau(1983) (resp. P. Kr\"{o}ger(1992)) found the lower (resp. upper) bound $\frac{d}{d+2}(2\pi)^2({\rm Vol}({\mathbb S}^{d-1}){\rm Vol}(\Omega))^{-2/d}k^{1+2/d}$ for the $k$-th (resp. ($k+1$)-th) eigenvalue of the Dirichlet (resp. Neumann) Laplacian. We show in this paper that this bound relates to the upper bound for $k$-th excited state energy eigenvalues of magnetic Schr\"{o}dinger operators with the compact resolvent. Moreover, we also investigate and mention the gap between two energies of particles on the magnetic field. For that purpose, we extend the results by Li, Yau and Kr\"{o}ger to the magnetic cases with no or Robin boundary conditions on the basis of their ideas and proofs.