Spectral theory in action

In this book, we present the basic tools of spectral analysis and illustrate the theory by presenting many examples from the theory of Schrödinger operators and from various branches of physics, including statistical mechanics, superconductivity, fluid mechanics, and kinetic theory. Hence we shall alternately present parts of the theory and use applications in those fields as examples. In the final chapters, we also give an introduction to the theory of non-self-adjoint operators with an emphasis on the role of pseudospectra. Throughout the book, the reader is assumed to have some elementary knowledge of Hilbertian and functional analysis and, for many examples and exercises, to have had some practice in distribution theory and Sobolev spaces. This introduction is intended to be a rather informal walk through some questions in spectral theory. We shall answer these questions mainly “by hand” using examples, with the aim of showing the need for a general theory to explain the results. Only in Chapter 2 will we start to give precise definitions and statements.

Our starting point is the theory of Hermitian matrices, that is, the theory of matrices satisfying A⋆ = A, where A⋆ is the adjoint matrix of A. When we are looking for eigenvectors and corresponding eigenvalues of A, that is, for pairs (u, λ) with u ∈ ℂk, u ≠ 0, and λ ∈ ℂ such that Au = λu, we know that the eigenvalues will be real and that one can find an orthonormal basis of eigenvectors associated with those eigenvalues. In this case, we can speak of eigenpairs.