Spectral geometry in a rotating frame: properties of the ground state

2019/02/08 by Barseghyan, Diana, Exner, Pavel
#35J15 #35P15 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1902.03038

Abstract

We investigate spectral properties of the operator describing a quantum particle confined to a planar domain Ω rotating around a fixed point with an angular velocity ω and demonstrate several properties of its principal eigenvalue λ₁^ω. We show that as a function of rotating center position it attains a unique maximum and has no other extrema provided the said position is unrestricted. Furthermore, we show that as a function ω, the eigenvalue attains a maximum at ω=0, unique unless Ω has a full rotational symmetry. Finally, we present an upper bound to the difference λ_1,Ω^ω- λ_1,B^ω where the last named eigenvalue corresponds to a disk of the same area as Ω.

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