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Scheda Riassuntiva
Anno Accademico 2023/2024
Tipo incarico Dottorato
Insegnamento 061603 - SPECTRAL AND SCATTERING THEORY IN QUANTUM MECHANICS
Cfu 5.00 Tipo insegnamento Monodisciplinare
Docenti: Titolare (Co-titolari) Borrelli William (Fermi Davide)

Corso di Dottorato Da (compreso) A (escluso) Insegnamento
MI (1385) - MODELLI E METODI MATEMATICI PER L'INGEGNERIA / MATHEMATICAL MODELS AND METHODS IN ENGINEERINGAZZZZ061603 - SPECTRAL AND SCATTERING THEORY IN QUANTUM MECHANICS

Programma dettagliato e risultati di apprendimento attesi

Description of the course

The course consists of two parts. General results for self-adjoint operators will

be presented, with particular emphasis on Schrödinger operators, corresponding to

quantum Hamiltonians of physical interest. We will first describe various approaches

to the description of self-adjoint extensions of symmetric operators. Then we shall

deal with different classifications of the spectrum of self-adjoint operators, presenting

some results about stability under perturbations. The relation with the associated

quantum dynamics will be presented. This subject will be further explored in the

second part of the course, dealing with quantum scattering problems. We will focus

mainly on the one-particle case, presenting the basic aspects of the theory, in both the

time-independent and time-dependent formulations.

 

The course is organized as follows.

 

First part (W. Borrelli - 12 hours, approx.)

(1) Self-adjoint extensions of symmetric operators: von Neumann theory and resolvent

techniques. Quadratic form methods.

(2) Applications to Schrödinger operators.

(3) (Time permitting) Magnetic Schrödinger operators.

(4) Classification of the spectrum of self-ajoint operators and quantum dynamics.

(5) Perturbation theory for self-adjoint operators: stability of the essential spectrum

and existence of discrete eigenvalues. Min-max principles for eigenvalues

of Schrödinger operators.

 

Second part (D. Fermi - 13 hours, approx.)

(1) A review of operator ideals: compact, Hilbert-Schmidt and trace class operators.

(2) Non-relativistic one-body scattering theory: characterization of asymptotically

free states, existence and completeness of wave operators, time-dependent and

stationary methods (Kato-Birman theory), the role of the singular spectrum,

generalized eigenfunctions and the Lippmann-Schwinger equation (Agmon theory

and the Limiting Absorption Principle).

(3) Definition of the scattering operator and its connection to the physical cross

section. Perturbative expansions: Born and Dyson series.

 

At the end of the course the student shall be able to

- analyze basic spectral properties of self-ajoint operators

- understand the mathematical foundations of scattering theory in non-relativistic quantum mechanics 

- rigorously formulate basic mathematical problems related to the study of non-relativistic quantum dynamics

 

Prerequisites.

Basic knowledge of functional analysis and operator theory in Hilbert spaces, Schwartz distributions and Sobolev spaces, an introductory course on quantum mechanics.


Note Sulla Modalità di valutazione

Learning evaluation will constist in a seminar on a research paper or a textbook chapter.


Intervallo di svolgimento dell'attività didattica
Data inizio
Data termine

Calendario testuale dell'attività didattica
Lectures will take place twice a week in Sala Consiglio (VII floor) at the Mathematics Department of Politecnico di Milano. The course will start on 20th February 2024 and will consist of 2h lectures that will take place on Tuesday and Wednesday from 10 am to 12 pm.

Bibliografia
Risorsa bibliografica obbligatoriaM. Reed, B. Simon, Methods of modern mathematical physics Vol. I-IV,, Editore: Academic Press, Anno edizione: 1980
Risorsa bibliografica obbligatoriaE. Prugovecki, Quantum Mechanics in Hilbert space, Editore: Academic Press, Anno edizione: 1981
Risorsa bibliografica obbligatoriaN. I. Akhiezer and I. M. Glazmann, Theory of linear operators in Hilbert spaces, Editore: Dover Publications, Anno edizione: 1992

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Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese

Note Docente
schedaincarico v. 1.9.7 / 1.9.7
Area Servizi ICT
28/05/2024