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

Abstract

We investigate spectral properties of the operator describing a quantum particle confined to a planar domain $\Omega$ rotating around a fixed point with an angular velocity $\omega$ and demonstrate several properties of its principal eigenvalue ${\lambda }_1^{\omega }$. 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 $\omega$, the eigenvalue attains a maximum at $\omega =0$, unique unless $\Omega$ has a full rotational symmetry. Finally, we present an upper bound to the difference ${\lambda }_{1,\Omega }^{\omega }-{\lambda }_{1,B}^{\omega }$ where the last named eigenvalue corresponds to a disk of the same area as $\Omega$.