# On the spectrum of the local $\mathbb{P}^2$ mirror curve

## TL;DR

This paper investigates the spectral properties of a quantum operator derived from the mirror curve of local P^2, a toric Calabi-Yau threefold, focusing on complex Planck's constant values.

## Contribution

It provides a detailed analysis of the spectrum of the quantum operator associated with the local P^2 mirror curve for complex Planck's constants.

## Key findings

- Spectral characteristics of the quantum operator are characterized.
- Insights into the quantum geometry of local P^2 are obtained.
- The study extends understanding to complex Planck's constant values.

## Abstract

We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's constant.