| Contact | C.V. (vitæ) | Enseignement | Recherche (research) |
|---|---|---|---|
| Spectral Theory |
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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
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 also benefit from 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).
Those who struggle with English language may gather useful information by combining the book by Haïm Brezis, Analyse fonctionnelle on functional analysis (but the English version is way richer) with the book by Mathieu Lewin, Théorie spectrale et mécanique quantique on the self-adjoint case and the parts on the linear Cauchy problems from either of the books by Haïm Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert or by Thierry Cazenave and Alain Haraux, Introduction aux problèmes d'évolution semi-linéaires for semigroup theory.
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.
The chapter provides basic terminology for linear operators. The material is largely covered in the literature, including in Taylor's Appendix and Davies' book aforementioned.
The chapter is an introduction to strongly continuous one-parameter semigroups of linear operators, including Hille-Yosida generating theorem and various reprensentation formulas both for general continuous semigroups and specific to analytic semigroups. The material is also covered by Davies' book aforementioned.
Yet students may benefit from consulting Arendt-Batty-Hieber-Neubrander's book aforementioned and/or the book by Amnon Pazy, Semigroups of Linear Operators and applications to partial differential equations.
For elementary general background on exponential integrals the students are referred to the book by Peter Miller, Applied asymptotic analysis.
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).
The homework is due for the course of October 13th. It must be hand-written, in either English or French language. This evaluation shall be completed by a short oral examination on the same material.