ADVANCED MATHEMATICAL METHODS FOR PHYSICS

The course aims to deal with some formalisms and mathematical tools of interest to modern physics, and in particular to quantum physics.

The students will obtain notion pertaining to algebra and linear operator theory of finite and infinite dimensional vector spaces, with a special focus on the spectral properties of operators. The couse will underline the deep connection with quantum mechanics. Some relevant concepts of measure theory will be also covered, as well as some equations of mathematical physics.
Such knowledge will allow a deeper comprehension of fundamental concepts, such as the self-adjointness of operators and their spectral properties.
The mathematical skills provided by the couse will allow a firmer foundation to the mathematical investigations of quantum physics, with the support of a more rigorous language and mathematical background.

Lectures

Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

Finite dimensional vector spaces, linear operators, eigenvalue problems. Recalls of measurement theory, L^p spaces. Euclidean spaces, and Hilbert space, orthonormal bases. Operators in Hilbert spaces. Fourier series and transform. Distributions. Spectral theory and methods for spectrum calculation. Some PDE of mathematical physics.

G. Fonte, Appunti di metodi matematici della fisica, Aracne
C. Rossetti, Metodi matematici per la Fisica, Levrotto & Bella.
G. Cicogna, Metodi matematici della Fisica, Springer.