Spectral equations for the modular oscillator

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.

23 pages, 9 figures, references added and interpretation of the numerical results of Section 6 corrected

References in corpus (2)

• Exact eigenfunctions and the open topological string
• Calabi-Yau geometry and electrons on 2d lattices

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• Solution of Baxter equation for the $q$-Toda and Toda${}_2$ chains by NLIE
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• On eigenstates for some $s{l}_2$ related Hamiltonian