2019/05/30 by Norihiro Someyama, Someyama, Norihiro
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1906.03257
openalex publication_date 2019/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider magnetic Schrödinger operators on a bounded region Ω with the smooth boundary ∂ Ω in Euclidean space \mathbb Rd. In reference to the result from Weyl's asymptotic law and Pólya's conjecture, P. Li and S. -T. Yau(1983) (resp. P. Kröger(1992)) found the lower (resp. upper) bound (d)/(d+2)(2π)2(\rm Vol(\mathbb Sd-1)\rm Vol(Ω))-2/dk1+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 Schrödinger 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 Kröger to the magnetic cases with no or Robin boundary conditions on the basis of their ideas and proofs.