Eigenvalue Problem Of Ginzburg-Laudan Operator With A Non-Smooth Magnetic Filed

We establish an asymptotic estimate of the lowest eigenvalue μ(σA) of the Schrodinger operator — â–½_σA² with a magnetic field in a bounded 2-dimensional domain, where A is a non-smooth magnetic potential, and σ is a large parameter. The case where A is smooth has been extensively studied recently. However when A is not smooth, little is known. Our study is based on an analysis on an eigenvalue variation problem for the associated Sturm-Liouville probelm. One of our main results is the following.Theorem 1. Let Ω be a smooth and bounded domain in R². There exists a universal constant β₀, 0 <β₀ < 1, such that for all A ∈ W^1,∞(Ω),hereWe also examine the lower bound of μ(σA) in the case where A is a C¹ vector field. The lowest eigenvalue of the Schrodinger operator in R² and in R₊² with A = (-|x₂|^α,0) is also investigated.