Provides a self-contained introduction to spectral theory while making important connections with practical problems. Covers the most important themes of this branch of mathematical physics so that the reader may easily move on independently to seek a more profound understanding of particular topics. Usually, self-contained texts of this sort are relatively large. On the other hand, those existing texts which are smaller are intended for a postgraduate audience where it is presumed that the reader is familiar with the multiple tools of Functional Analysis. Both of these approaches tend to overload and distract students with all kinds of little things that might only be of occasional use. An Introduction to Spectral Theory is an effort to strike a balance between these extremes. Many of the theorems are stated in less than maximal generality, which makes the essential ideas more accessible. The topics of Functional Analysis are presented insofar as they are necessary. However, the reader will find self-sufficient proofs of the results concerning linear functionals acting in Hilbert spaces, L_p and C, weak and strong compactness in L_p, the Fredholm equations, the Fourier transform, the weak partial derivatives, the Sobolev spaces, the point, continuous and essential spectra of linear operators, the weak and singular solutions of differential equations, etc.

The book starts with the consideration of the compact sets in different functional spaces, including Banach spaces with basis, L_p and C, which is followed by the spectral decomposition of compact self-adjoint operators in Hilbert spaces. The first chapter is introductory and can be omitted by readers who are familiar with basic topics of Functional Analysis. Chapters 2 and 3 are devoted to the symbiosis of the theory of compact operators in Hilbert spaces and the study of boundary-value problems for elliptic differential equations. The Fredholm theorems are proved and the details of the reduction of the Sturm-Liouville problem to an integral equation are discussed. Chapter 4 is dedicated to the integral representation of self-adjoint operators in the form of Stieltjes integral with respect to the spectral measure. Chapters 5 and 6 deal with the explicit spectral decomposition of the Laplacian in L2 and the study of the structure of the Laplacian acting in the whole space. Chapter 7 studies eigenvalues of the Laplacian in bounded domains, making more precise the co-existence of compact operators and boundary value problems. The results of Chapter 8 are relatively new. This chapter is devoted to the spectrum of the operators generated by systems of different types of rotating fluid, both in bounded domains and the whole space. The Weyl sequence is constructed, which enables not only the establishment of the spectral properties, but also the obtainment of explicit solutions of the considered systems.

Prerequisites for the book are: a very limited knowledge of Hibert and Banach space definitions, an understanding of the definition of the Lebesque integral, and a standard course on ordinary differential equations.