# Spectral equations for the modular oscillator

## TL;DR

This paper investigates the spectral problem of a pair of modular Harper operators related to non-perturbative topological strings, providing analytical insights supported by numerical calculations.

## Contribution

It introduces a spectral analysis of modular Harper operators linked to quantized mirror curves in topological string theory, a novel connection in the field.

## Key findings

- Analytical solutions for the spectral problem.
- Numerical validation of theoretical results.
- Connection between spectral equations and topological string applications.

## Abstract

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations.