In this chapter, we study the one-dimensional Schrödinger operator \(\begin{aligned} L(Q)=-\frac{d^{2}}{dx^{2}}+Q \end{aligned}\) acting in the space \(L_{2}^{m}(-\infty ,\infty )\) , where \(m\ge 2\) , \(Q=\left( q_{i,j}\right) \) is an \(m\times m\) matrix with the complex-valued locally summable entries \(q_{i,j}\) , such that \(q_{i,j}\left( x+1\right) =q_{i,j}\left( x\right) \) almost everywhere. This chapter consists of seven sections. The first section is introductory, where we introduce the necessary notations and concepts. In the second section, we study the localization of the Bloch eigenvaluesBloch eigenvalue. In the third section, asymptotic formulasAsymptotic formulas for the Bloch eigenvaluesBloch eigenvalue and eigenfunctions are obtained. In fourth and fifth sections, using these asymptotic formulasAsymptotic formulas, the spectral expansion for the operator L(Q) is constructed. In the sixth section, we investigate the Bloch eigenvaluesBloch eigenvalue in detail, and in the seventh section, building on these investigations, we consider the spectrum of L(Q) when the entries \(q_{i,j}\) of Q are PT-symmetric periodic functions.