ABSTRACT
This chapter aims to make a quick introduction to the basics of the spectral theory of differential operators and presents the basics of the general theory in the case of bounded and unbounded self-adjoint operators. The spectral theory of operators in infinite-dimensional spaces is much richer: while the spectrum of finite-dimensional operators consists only of eigenvalues, in the infinite-dimensional spaces we have other variants of the spectrum. The chapter discusses the simplest properties of the spectrum for linear bounded operators. The spectral theory of compact operators plays an important role in the spectral theory of differential operators. This is due to the fact that the operator inverses to the differential operators are compact in many cases. One of the simplest ordinary differential operators which is frequently encountered in applications to many areas of different sciences is the Sturm-Liouville operator.
