  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.
