In this chapter, we discuss some preliminary facts that will help to read the following chapters and grasp the purpose and value of this book. First, we explain the importance of spectral theorems for self-adjoint operators in Quantum Mechanics. Next the differences in methods for studying self-adjoint and non-self-adjoint operators are discussed. Additionally, we explain the need to search for new methods for various cases of the non-self-adjoint Schrödinger operators. Finally, we discuss the method and difficulties of studying the Schrödinger operatorSchrödinger operator with a periodic complex-valued potentialComplex-valued potential.