Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part I : For the Weber Equation

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part [IKoT] establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article, and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers.