Miguel Rodrigues

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Spectral Theory

The course is devoted to some aspects of the theory of unbounded linear operators. It includes introductions to semigroups of linear operators (as involved in the study of linear evolution equations) and spectral theory (notably the spectral theorem for self-adjoint operators). Most illustrations are taken from the study of linear partial differential equations.

The content of the course is essentially covered by

Jointly with Michael E. Taylor, Partial Differential Equations III: Nonlinear Equations, those two books form a trilogy. Other chapters of the trilogy may be used for background and also cover a large part of other courses on other aspects of the analysis of partial differential equations, and on differential geometry and stochastic analysis.

Other possible, narrower but single-book, references are either E. Brian Davies, Linear Operators and their Spectra or Christophe Cheverry and Nicolas Raymond, A Guide to Spectral Theory.

Those who are purely interested by linear analysis and applications to quantum mechanics may prefer to use the, older, quadrilogy by Michael Reed and Barry Simon, Methods of Modern Mathematical Physics (I Functional Analysis; II Fourier Analysis, Self-Adjointness; III Scattering Theory; IV Analysis of Operators).

Outline

Infinite-Dimensional Valued Functions

This chapter is a survival kit for infinite-dimensional analysis (topology, integration, differentiation, holomorphy). The material of this chapter is detailed in Chapter 1 of Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber and Frank Neubrander, Vector-valued Laplace Transforms and Cauchy Problems.

Unbounded Operators

The chapter provides basic terminology for linear operators.

Tests

The course includes two tests: one homework completed by an oral examination (CC1), one final in-class evaluation (CC2).

The final course grade is then obtained through max((CC1+CC2)/2,CC2).