Lower bounds for the first eigenvalue of the magnetic Laplacian

Abstract

We consider a Riemannian cylinder Ω endowed with a closed potential 1-form A and study the magnetic Laplacian ∆A with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.

Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

“…In fact, Balinsky uses a conformal mapping argument to obtain a lower bound for doubly connected domains in terms of conformal geometry. This bound is similar in the spirit to the lower bound in [4], which we use in our proof here, and which is expressed in terms of explicit geometric quantities.…”

“…The strategy of the proof is to partition the given punctured domain in a family of convex domains with only one puncture and then to apply a lower bound proved in [4] to each piece of the partition.…”

“…The referee also pointed out to us the interesting paper [9], where the authors give sharp lower bounds for punctured disks. However, since in our proof, the domains are not punctured disks but punctured convex sets, we have prefered to use our lower bound in [4].…”

“…Let Γ( ) be the -neighborhood of Γ and set D = Ω ⧵ Γ( ) ; if is small enough D is simply connected and we have, by Proposition 12 in [4]:…”

“…In our previous papers [4] and [5], we obtained lower bounds for the ground state energy. In [4], we proved an estimate from below of the first eigenvalue of a Riemannian cylinder; applied to a plane annulus Ω = F ⧵Ḡ , with F and G smooth and convex, the lower bound becomes:…”