Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

In this paper we construct the quantum spectral curve for the quantum dynamical elliptic $\mathfrak{gl}_n$ Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group $E_{\tau,\hbar}(\mathfrak{gl}_n)$ and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.