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

## TL;DR

This paper extends classical bounds on eigenvalues of Laplacians to magnetic Schr"{o}dinger operators with no or Robin boundary conditions, analyzing eigenvalue gaps and their relation to magnetic fields.

## Contribution

It introduces new bounds for magnetic Schr"{o}dinger operators' eigenvalues and explores energy gaps, extending prior results to magnetic and Robin boundary cases.

## Key findings

- Established bounds for magnetic Schr"{o}dinger eigenvalues.
- Analyzed energy gaps between particle states in magnetic fields.
- Extended classical eigenvalue bounds to magnetic and 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.