Niels Benedikter

Academic Year 2023-2024

PhD course: Mathematical Methods for Many-Body Quantum Systems, Milan, Fall 2023

After a very brief review of the mathematical framework of quantum mechanics, I will discuss second quantization methods for the analysis of many-body quantum systems. Topics include Fock space, creation and annihilation operators, coherent states, Bogoliubov transformations, quasifree states, Bose-Einstein condensation, and if time permits touch also upon aspects of variational approximations such as Hartree-Fock theory and the BCS theory of superconductivity.

Evaluation: Oral exam, pass/fail. Contact me by e-mail to make an appointment, available period: 4 - 8 March 2024. 3 CFU, in any form that is compatible with your study plan.

Place: Aula dottorato, first floor, Via Saldini 50.

  • Wednesday, 8 November 2023, 13:30 (sharp) – 16:05

    Hilbert spaces, densely defined operators, compact operators, trace-class and Hilbert-Schmidt operators, integral kernels

  • Wednesday, 15 November 2023, 13:30 – 16:05

    spectrum and Weyl criterion, confined systems, thermodynamic limit, tensor product of Hilbert spaces, unitary representation of the symmetric group, symmetric and antisymmetric tensor product, Slater determinants, Hamiltonians with pair interaction

  • Wednesday, 22 November 2023, 12:45 – 15:25

    Fock space, creation and annihilation operators, canonical (anti)commutation relations

  • Wednesday, 29 November 2023, 13:30 – 16:05

    second quantization, reduced density matrix, pairing density, generalized 1-pdm, generalized creation/annihilation operators, Bogoliubov transformations, quasifree states

  • Wednesday, 13 December 2023, 13:30 – 16:05

    generalized Hartree-Fock theory, phonon field, Fröhlich Hamiltonian, first order perturbation theory, phonon-mediated effective interaction between electrons, rigorous state of the art in the theory of superconductivity

  • Exercises for Exam Preparation

    You are welcome to come to my office or write an e-mail to ask questions about the lecture or the exercises.

Literature:

  • Jan Philip Solovej: Many Body Quantum Mechanics, Draft of Lecture Notes of March 5, 2014
  • Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volume I.
  • E. Kolley and W. Kolley: Derivation of Attractive Electron-Electron Interactions for Superconductivity via Canonical Transformations. Annalen der Physik, Volume 502, Issue 2-3, Pages 150-156 (1990). DOI: 10.1002/andp.19905020209
Metodi matematici della meccanica quantistica, Milan, Fall 2023

The first part will be similar to the course of Advanced Mathematical Physics I taught 2017 in Copenhagen (see below for lecture notes covering approximately two thirds of the present course). I will introduce von Neumann's framework of unbounded operators in separable Hilbert spaces as a rigorous base for quantum mechanics. We will then discuss Schrödinger operators (H = - ∆ + V) in this framework and analyze physical questions such as existence of solutions to the time-dependent Schrödinger equation, symmetries, time-dependent scattering theory, exponential localization of bound states, stability of the essential spectrum, as well as perturbation theory and its convergence.

Evaluation: To be admitted to the oral exam, reach approximately 50% of the points averaged over the homework assignments, to be handed in every two weeks for correction. Oral exam, no written exam. In the period 26 - 29 February 2024. Write me a mail to make an appointment.

Lecture calendar and exercise sheets:

  • Wednesday, 27 September: the formalism of quantum mechanics, Banach spaces, Lp-spaces, bounded operators, Hilbert spaces, Fréchet-Riesz representation theorem, closed graph theorem, densely defined operators
  • Friday, 29 September: resolvent, spectrum, analytic functions, operator-valued analytic functions, Neumann series
  • Wednesday, 4 October: operator adjoint, symmetric and self-adjoint operators, Hellinger-Töplitz theorem, momentum operator, generalized theorem of the bounded inverse, self-adjointness criterion
  • Friday, 6 October: exercise solutions and discussion
  • Wednesday, 11 October: self-adjointness criterion (cnt.), Kato-Rellich theorem, Schrödinger equation, essential self-adjointness, Schrödinger operators, unitary equivalence, Fourier transform
  • Friday, 13 October: multiplication operators, Weyl criterion, Laplace operators, Sobolev inequalities
  • Wednesday, 18 October: uniform boundedness principle, strong convergence, strongly continuous unitary groups, generator of a SCUG
  • Friday, 20 October: exercise solutions and discussion
  • Wednesday, 25 October: existence of SCUGs, commuting operators, conserved quantities, symmetries, Quantum Noether theorem
  • Friday, 27 October: scattering theory I: in- and out going states, wave operators, bound states, scattering states
  • Wednesday, 8 November: scattering theory II: S-matrix, asymptotic completeness in absence of bound states
  • Friday, 10 November: exercise solutions, review on residues, review on Riemann-Lebesgue lemma
  • Wednesday, 15 November: stationary scattering theory, IMS formula, localization of bound states
  • Friday, 17 November: Sobolev lemma, regularity of bound states, compact operators
    • Exercises: Sheet 4 to be handed in Wednesday, November 22
  • Wednesday, 22 November: Hilbert-Schmidt operators, relative compactness
  • Friday, 24 November: exercise solutions
  • Wednesday, 29 November: stability of the essential spectrum, functional calculus I
  • Friday, 1 December: functional calculus II
    • Exercises: Sheet 5 to be handed in Wednesday, December 13
  • Wednesday, 6 December: no lecture
  • Friday, 8 December: no lecture
  • Wednesday, 13 December: spectral theorem
  • Friday, 15 December: spectral subspaces (1h) - exercise solutions (1h)
    • Exercises: Sheet 6 to be handed in Wednesday, January 17
  • Wednesday, 20 December: no lecture
  • Friday, 22 December: no lecture
  • Wednesday, 10 January: no lecture
  • Friday, 12 January: no lecture
  • Wednesday, 17 January: absolutely continuous and singularly continuous spectrum, Wiener's theorem, RAGE theorem (abstract)
  • Friday, 19 January: RAGE theorem (Schrödinger operators) - solutions sheet 6 - END

Literature:

  • Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version

    Contains essentially all topics of the lecture, but in a different arrangement. Some details are missing.

  • Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version

    Nice selection of material, but not all proofs are given completely. Good overview.

  • Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.

    Recommended as a reference work, not as a text book. Concise but rather dense.

  • Elliott H. Lieb, Michael Loss: Analysis. Errata

    A functional analysis book with a constructive point of view. Contains many useful explicit estimates.

If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):

  • Gordon Baym: Lectures on Quantum Mechanics
  • Leslie E. Ballentine: Quantum Mechanics, A Modern Development
  • Steven Weinberg: Lectures on Quantum Mechanics
Fisica Matematica 3 (Meccanica Quantistica), Milan, Spring 2024

Seconda parte del corso di Fisica Matematica 3. (Prima parte Meccanica Statistica con il Prof. Vieri Mastropietro.)

Academic Year 2022-2023

Sistemi Hamiltoniani laboratorio, Milan, Fall 2022

There are partial notes to be expanded over time.

Matematica generale (Biologia), Milan, Fall 2022

Informazioni su Ariel.

Meccanica Analitica (Fisica), Milan, Fall 2022

Informazioni su Ariel.

Metodi e modelli matematici per le applicazioni laboratorio, Milan, Fall 2022

Informazioni su Ariel.

Academic Year 2021-2022

Fisica Matematica 3 (Meccanica Statistica), Milan, Spring 2022

Videoregistrazione su Ariel. Riassunti delle lezioni:

Libri ed altre risorse:

  • Teitel, Stephen: Statistical Mechanics I (Lecture notes 2021), University of Rochester
  • Porta, Marcello: Mathematical aspects of classical statistical mechanics (Lecture notes 2015), University of Zurich -> Details -> Script
  • Gros, Claudius: Thermodynamik & Statistische Mechanik (Lecture notes 2017/2018), University of Frankfurt
  • Schwabl, Franz: Statistical Mechanics, Springer 2006, ISBN 978-3-642-06887-4
  • Ruelle, David: Statistical Mechanics - Rigorous Results, World Scientific 1999, ISBN 978-9-810-23862-9
  • Kiessling, Michael: On Ruelle’s construction of the thermodynamic limit for the classical microcanonical entropy, arXiv:0810.2084, J. Stat. Phys. 134, pp.19-25 (2009)
  • Luca Peliti: Statistical Mechanics in a Nutshell, Princeton University Press 2011, ISBN 978-0-691-20177-1
  • Sascha Friedli and Yvan Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press 2017, ISBN 978-1-107-18482-4
Sistemi Hamiltoniani laboratorio, Milan, Fall 2021

Informazioni su Ariel.

Academic Year 2020-2021

Sistemi Hamiltoniani laboratorio, Milan 2020/2021

This course deals with Hamiltonian mechanics and their perturbation theory. Recordings for my part of the theory lectures (corresponding to Chapter 2 "Canonical Transformations" of the lecture notes) can be found on Ariel (link below).

For materials and recording of the associated programming course, please see the official Ariel page. Every section on Ariel contains a subsection "Lezioni" and "Laboratorio"; for the programming course go to "Laboratorio". Here is the overview of the topics I discussed up to now:

  • Lab 1: Introduction to Linux, Installing Mizar, Introduction to C
  • Lab 2: Working with arrays, memcpy, and a first idea of numerics: solving a variational problem
  • Lab 3: Mizar and the forward Euler method
  • Lab 4: Symplectic integrators: the Leapfrog method
  • Lab 5: analysis of harmonic oscillators with linear coupling, Poincare sections
  • Lab 6: Henon-Heiles model, algorithm for Poincare sections [Henon1982], analytic solution for linearly coupled HOs
  • Lab 7: Splitting methods from Leapfrog to SABA3
  • Lab 8: SABA3 with corrector and "smarter" splitting; introduction to construction of first integrals
  • Lab 9: conclusion numerics, algebraic computation with polynomials in one variable
  • Lab 10: algebraic computation with polynomials in multiple variables, Poisson bracket in complex representation
  • Lab 11: iterative construction of first integrals for the Hénon-Heiles model
  • Lab 12: analysis of algebraically constructed truncated first integrals along numerical trajectories
  • Lab 13: plotting level sets of truncated first integrals
  • Lab 14: comparison of level sets to Poincare sections
Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2020/2021

Motivazione per la matematica: "This equation will change how you see the world (the logistic map)".

Insegnerò solo il secondo semestre, da aprile 2021. La prima parte (fino a fine marzo) è tenuta da Prof. Maggis. Per informazioni guardate la pagina Ariel del corso.

I teach only the second semester, from April 2021. The first part (until end of march) is taught by Prof. Maggis. For more information, please look at the Ariel page of the course.

Youmath.it: Essercizi aggiuntivi e lezioni.

Academic Year 2019-2020

Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2019/2020

L'insegnamento fornisce gli strumenti base dell'Analisi Matematica, sia dal punto di vista teorico che pratico, indispensabili per poter seguire con profitto un corso universitario di carattere scientifico. Le conoscenze proposte sono propedeutiche ad altri corsi base del CdS.

Descrizione sulla pagina dell'Università: Matematica del continuo. Tutte le informazioni su Ariel. Per ora (per il corona virus) le lezioni sono su Zoom in orario normale (martedì 10:30-13:30, mercoledì 8:30-11:30).

Riassunto di quasi tutto: riassunto per stampare. Attenzione: non contiene le lezioni di Lorenzo Luperi Baglini (in particolare il capitolo sul integrale definito), quelli si trovano solo su Ariel. Invece, tutti i riassunti dell'elenco seguente sono contenuti nel file mdc2up.pdf.

Riassunti (anche su Ariel):

Esercizi (anche su Ariel):

Before 2019

Teaching assistant, IST Austria 2018

Teaching assistant for the course "Stability of Matter in Quantum Mechanics" with Prof. Robert Seiringer.

Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. This course provides a self-contained description of research on the stability of matter problem. It introduces the necessary aspects of functional analysis as well as the quantum mechanical background. The topics covered include Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, and gravitational stability of stars.

Advanced Mathematical Physics, Copenhagen 2017

Lecture Notes:

Lectures Niels:

Lectures Jérémy:

Assignments (to be handed in at the beginning of the Friday seminar!):

Summer School on Current Topics in Mathematic Physics at the University of Zurich, Switzerland, July 17 - July 21:

Seminar: All talks have to be about 40 minutes long (not more!). Schedule is subject to changes depending on our progression in the lecture. The summary is due on Monday after the seminar. Please remember to provide a list of references. Contact Jérémy or me at least two weeks before your talk for a briefing. Topics and summaries:

Criteria for passing the course:
Reach approximately 50% of the points averaged over the four assignments.
Give a seminar talk and produce a summary of your talk for the other participants.

Literature: The lecture notes should be self-contained for most of the course. If you are looking for additional reading, here are some recommendations.

  • Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version

    Contains essentially all topics of the lecture, but in a different arrangement. Sometimes details are missing.

  • Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version

    Nice selection of material, but sometimes sketchy. Good overview.

  • Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.

    Recommended as a reference work, not as a text book. Concise but rather dense.

  • Elliott H. Lieb, Michael Loss: Analysis. Errata

    A functional analysis book aimed at applications in quantum mechanics. Contains many useful explicit estimates.

If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):

  • Gordon Baym: Lectures on Quantum Mechanics
  • Leslie E. Ballentine: Quantum Mechanics, A Modern Development
  • Steven Weinberg: Lectures on Quantum Mechanics
Advanced Mathematical Physics, Copenhagen 2016

Thank you all for following the course! New materials for the 2017 course will appear above.

Teaching Assistance

Teaching assistant, Bonn 2011-2014

I assisted Prof. Benjamin Schlein at the University of Bonn for the courses

and for a Summer School at the University of Heidelberg