In infinite dimension, the spectral theory of self-adjoint operators is much more subtle than for hermitian matrices. Many physical or mechanical practical problems can be formulated as an eigenvalue equation whose unknown is a function, or as a partial differential equation which can be studied with spectral methods.
Developed by Hilbert at the end of the 19th century, spectral theory has really emerged in the 20-30s after the invention of quantum mechanics and of Schrödinger's equation, in particular with works by Stone and von Neumann.
In this course we describe the basic spectral theory of self-adjoint operators in infinite dimension and we apply it to several quantum mechanical systems, including for instance atoms and molecules.
Bibliography
Lecture notes in French
Lewin, Spectral theory and quantum mechanics, Universitext, Springer International Publishing, 2024
Davies, Spectral theory and differential operators, Cambridge Univ. Press, 1995.
Reed, B. Simon, /Methods of Modern Mathematical Physics. Volumes I, II et IV/. Academic Press, 1978
Required level
- elements of Fourier analysis and distribution theory
- prior knowledges in quantum mechanics may help but are not mandatory to take this course
Course language: French
Credits ECTS : 5
- Teaching coordinator: Lewin Mathieu