### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
CORNELIS VICTOR MARIA VAN DER MEE (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/00 - Ord. 2012]  PERCORSO COMUNE 9 72
[60/68]  PHYSICS [68/00 - Ord. 2014]  PERCORSO COMUNE 9 72

### Objectives

To acquire an operational knowledge of the partial differential equations of stationary or evolution type relevant to mathematical physics. For this reason some of the arguments are illustrated with examples and exercises without formal distinction between the two.

### Prerequisites

The course requires a good knowledge of the basic concepts of mathematical analysis and linear algebra.

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### Contents

1. EQUATIONS OF MATHEMATICAL PHYSICS
a. Orthogonal corrdinates and Laplacian in orthogonal coordinates, in
particular in polar, cylindrical and spherical coordinates
b. Separation of variables in cartesian, polar, cylindrical and spherical
coordinates
c. Helmholtz, heat and wave equations on the interval and in a rectangle

2. FUNCTIONAL ANALYSIS
a. Banach and Hilbert spaces
b. Orthonormal basis, Gram-Schmidt orthonormalization, application to Fourier
series
c. Bounded linear operators, their spectra, selfadjoint and unitary operators

3. DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS
a. Second order differential equations: Existence and uniqueness of solutions,
linear (in)dependence and Wronskian, method of variation of parameters
b. Solution by means of expansion in power series: Frobenius method
c. Hypergeometric and confluent hypergeometric functions
d. Bessel functions: Power series, behavior in zero and at infinity, zeros,
orthogonality, Neumann and Hankel functions, imaginary Bessel functions.
e. Spherical functions: definition, orthogonality, Laplace-Beltrami operator,
in 2D and 3Df. Legendre and associate Legendre functions: Rodriguez formula, differential
equation, orthogonality, recurrence relation, normalization
g. Classical orthogonal polynomials: Chebyshev, Hermite, Laguerre
h. General orthogonal polynomials: zeros (number, multiplicity, position,
interlacing)
i. Gamma function

4. INTEGRAL EQUATIONS
a. Boundedness of integral operators in L1, L2 and C
b. Volterra integral equations
c. Rayleigh-Ritz principle for hermitian integral equations
d. Hilbert-Schmidt theorem

5. STURM-LIOUVILLE PROBLEMS
a. One-dimensional Sturm-Liouville problems: Boundary conditions, properties of
the eigenvalues, examples
b. Conversion into an integral equation and one-dimensional Green functions

6. GREEN FUNCTIONS
a. Classification of second order partial differential equations
b. Multidimensional Sturm-Liouville problems: boundary conditions, properties
of the eigenvalues c. Laplace-Poisson equations (in intervals, on the entire space, on the
half-plane, on the disk, on the sphere) in 2D and 3D
d. Helmholtz equation
e. Heat equation
f. Wave equation, D'Alembert formula
g. Distributional solutions

7. SCHROEDINGER EQUATIONS
a. Radial Schroedinger equation: potential gap, oscillator, hydrogen
b. Schroedinger equation with periodic potential

### Teaching Methods

Classroom lectures (using blackboard and/or projector) and problem sessions, with full integration of the two.

### Verification of learning

The evaluation is based on an oral exam preceded by a written test, evaluated as a package, covering the main arguments of the course.

### Texts

Lecture notes available online. These lecture notes contain a bibliography which the student may consult.