O. A. Veliev The Periodic Multidimensional Schrodinger Operator, Part 1, Asymptotic Formulae for Eigenvalues (83K, LATeX 2e) ABSTRACT. In this paper we obtain asymptotic formulas of arbitrary order for the eigenvalues of the $d$-dimensional periodic Schrodinger operator $% L(q)=-\Delta +q(x)$ for arbitrary dimension $d$. The eigenvalues, whose asymptotic formulas are connected with Sturm-Liouville operators, are considered in detail.