  This paper is devoted to the eigenvalue problem for the quantum Gaudin
system. We prove the universal correspondence between eigenvalues of Gaudin
Hamiltonians and the so-called G-opers without monodromy in general gl(n) case
modulo a hypothesys on the analytic properties of the solution of a KZ-type
equation.
  Firstly we explore the quantum analog of the characteristic polynomial which
is a differential operator in a variable $u$ with the coefficients in
U(gl(n))^{\otimes N}. We will call it "universal G-oper". It is constructed by
the formula "Det"(L(u)-\partial_u) where L(u) is the quantum Lax operator for
the Gaudin model and "Det" is appropriate definition of the determinant. The
coefficients of this differential operator are quantum Gaudin Hamiltonians
obtained by one of the authors (D.T. hep-th/0404153). We establish the
correspondence between eigenvalues and $G$-opers as follows: taking
eigen-values of the Gaudin's hamiltonians on the joint eigen-vector in the
tensor product of finite-dimensional representation of gl(n) and substituting
them into the universal G-oper we obtain the scalar differential operator
(scalar G-oper) which conjecturally does not have monodromy. We strongly
believe that our quantization of the Gaudin model coincides with quantization
obtained from the center of universal enveloping algebra on the critical level
and that our scalar G-oper coincides with the G-oper obtained by the geometric
Langlands correspondence, hence it provides very simple and explicit map
(Langlands correspondence) from Hitchin D-modules to G-opers in the case of
rational base curves. It seems to be easy to generalize the constructions to
the case of other semisimple Lie algebras and models like XYZ.
