# GUIDELINES OF MATHEMATICAL METHODS OF PHYSICS

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Versione italiana
2021/2022
Teacher
GIUSEPPE PAGLIARA
Credits
9
Didactic period
Secondo Semestre
SSD
FIS/02

#### Training objectives

The main aim of this course is to provide the students with the most advanced mathematical tools which are necessary to face the study of the modern physics courses of the third year. In particular, the mathematical foundations of quantum mechanics will be presented with a particular emphasis of the notion of Hilbert space. The student will be able to think in abstract linear spaces with infinite dimensions.
The other important aim of this course is to make the students able to handle the advanced techniques of complex analysis and of Fourier transforms. These will be very useful for the subsequent studies of both theoretical and experimental physics.

#### Prerequisites

The mathematical methods and techniques which have been introduced in the courses of mathematical analysis (differential and integral calculus in one or more dimensions, differential equations) and in the course of geometry (linear algebra, analytical geometry).
To participate to the exam sessions it is mandatory to pass the exam of Mathematical Analysis I.

#### Course programme

The course is divided into three main parts: complex analysis, theory of linear spaces with infinite dimensions (Hilbert spaces in particular), Fourier transform and theory of distributions.

1. Complex Analysis (28 hours): reminder of complex numbers (2h). Analitical functions, series, Taylor - Laurent series (4h). Integral in the complex plane and Cauchy theorems (4h). Roots and singularities (2h).Residue theorem, Cauchy principal value and Jordan's lemmas (8h). The point and infinity and its residue (2h). Roots and logaritm, branch points and cuts (6h).

2.Hilbert spaces (28h): reminder of linear algebra (finite dimensions, 2h). The equation of vibrating rope and the need of linear spaces in infinite dimensions, scalar product, norm, Banach and Hilbert spaces, L2 and L2 functional spaces and l2 space (10h). Linear operators, continuous and limited operators, hermitian and unitary operators (8h). The Sturm-Liouville problem and application to harmonic oscillator and hydrogen atom, special polynomials (4).
Diagonalization of self-adjoint operators, discrete and continuous spectrum (4h).

3. Fourier transform and theory of distributions (16 hours):
Fourier transforms in L1 and L2, inverse Fourier transform, Green function (10h). Elements of theory of distributions, convergence in S and S1, the delta, Heaviside and principal value distributions. Derivative and Fourier transform of distributions (6h).

4. Rigged Hilbert space and spectral theorem (4h).

Additional topics (from a.a. 2020-21): harmonic functions and analitical functions, Poisson formula, differential equations and Fuchs theorem, Mittag-Leffler expansions, elements on asymtpotic series (5h).

#### Didactic methods

Lectures are presented on the black-board to allow the students to follow the details of the calculations and of the mathematical proofs and to eventually ask questions already during the lecture. Some exercises will be shown in which one makes use of the software Mathematica.

#### Learning assessment procedures

The written exam is composed by four exercises: a first exercise on the properties of function of complex variable (singularities, series expansions, Cauchy integral formulae); a second exercise which is an integral to be computed with the methods of complex analysis (residue's method, functions with branch cuts); an exercise on Hilbert spaces (complete sets, linear operators and diagonalization, spectral theorem); the fourth exercise is about Fourier transform and distributions with possible application to differential equations. To pass the written exam one needs to reach at least a grade of 15. Books and notes can be used during the exam. This part of the exam is aimed at verifying if the student is able to deal with analitic functions and if is able to handle Hilbert spaces and linear operators.
The oral exam aims at testing the knowledge of the student of the most formal part of the course. Three questions will be asked concerning the three main blocks of the course. In particular theorems, examples and simple proofs can be asked during the exam.

#### Reference texts

Most of the course is based on the book
"Metodi matematici della fisica" di G. Cicogna. (editore Springer Verlag). The following chapters will be presented: cap. 3 (3.1-3.13). cap. 1 (1.1-1.8). cap.2 (2.1-2.4, 2.6-2.14, 2.16-2.17, 2.19-2.21, 2.23), cap. 4 (4.3-4.8). cap.5 (5.1-5.5,5.7).
Same author: "Compiti con Soluzioni di Metodi Matematici della Fisica".

Also the following book is suggested:
"A Guide to Mathematical Methods for Physicists: With Problems and Solutions" di Michela Petrini, Gianfranco Alberto Zaffaroni (editor World Scientific 2018).

For some specific topics the following books are recommended:
"Metodi Matematici della Fisica" di C. Bernardini, O. Ragnisco, P. M. Santini (Carocci editore), concerning Hilbert spaces;
"Metodi Matematici della Fisica" di C. Rossetti
(editore Levrotto & Bella) concerning complex analysis.