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

## TL;DR

This paper analyzes how the principal eigenvalue of a rotating quantum particle in a planar domain varies with the rotation center and angular velocity, revealing unique extremal properties and bounds related to domain shape.

## Contribution

It provides new insights into the spectral behavior of quantum particles in rotating frames, including extremal properties of eigenvalues and bounds related to domain geometry.

## Key findings

- Eigenvalue attains a unique maximum as a function of rotation center position.
- Eigenvalue attains a maximum at zero angular velocity unless the domain is rotationally symmetric.
- An upper bound is established for the eigenvalue difference between the domain and a disk of the same area.

## 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 $\Omega$.