Niels Benedikter

Updated April 10, 2025

Academic Year 2024-2025

Fisica Matematica 3 (Meccanica Statistica), Milan, Spring 2025

Il corso segue principalmente le note della versione 2022 dello stesso corso, vedi sotto.

Calendario delle lezioni:

  • 24/02/2025: cos'è la meccanica statistica, la teoria microscopica, l'ipotesi ergodica
  • 25/02/2025: l'insieme microcanonico, le quantità termodinamiche appunti aggiuntivi
  • 26/02/2025: il gas ideale
  • 27/02/2025: la correzione di Gibbs/entropia di miscelazione, compiti per 5/3/2025
  • 03/03/2025: l'insieme canonico, la distribuzione di Maxwell-Boltzmann
  • 04/03/2025: le quantità termodinamiche nell'insieme canonico
  • 05/03/2025: soluzione compito, trasformata di Legendre
  • 06/03/2025: il limite termodinamico
  • 10/03/2025: esistenza del limite termodinamico
  • 11/03/2025: esistenza del limite termodinamico
  • 12/03/2025: compito in presenza
  • 13/03/2025: esistenza del limite termodinamico
  • 17/03/2025: limite termodinamico
  • 18/03/2025: limite termodinamico dell'entropia
  • 19/03/2025: transizioni di fase
  • 20/03/2025: teoria di Van der Waals
  • 24/03/2025: gas su reticolo
  • 25/03/2025: modello di Ising in dimensione d = 1
  • 26/03/2025: modello di Ising: disuguaglianze di correlazioni
  • 27/03/2025: modello di Ising: disuguaglianze di correlazioni
  • 31/03/2025: modello di Ising: disuguaglianze di correlazioni
  • 01/04/2025: espansione temperatura bassa
  • 02/04/2025: soluzione compito + argomento di Peierls
  • 03/04/2025: argomento di Peierls + compito in presenza
  • MECCANICA QUANTISTICA
  • 04/06/2025: meccanica statistica di sistemi quantistici
  • 05/06/2025: compito in presenza (possibili argomenti: gas interagente, modello di Ising, meccanica statistica di sistemi quantistici non-interagenti)

Libri ed altre risorse:

  • Teitel, Stephen: Statistical Mechanics I (Lecture notes 2021), University of Rochester
  • 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)
  • Sascha Friedli and Yvan Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press 2017, ISBN 978-1-107-18482-4
  • Luca Peliti: Statistical Mechanics in a Nutshell, Princeton University Press 2011, ISBN 978-0-691-20177-1
  • Schwabl, Franz: Statistical Mechanics, Springer 2006, ISBN 978-3-642-06887-4
  • Gros, Claudius: Thermodynamik & Statistische Mechanik (Lecture notes 2017/2018), University of Frankfurt
  • Porta, Marcello: Mathematical aspects of classical statistical mechanics (Lecture notes 2015), University of Zurich -> Details -> Script
Metodi matematici della meccanica quantistica, Milan, Fall 2024

I will introduce von Neumann's framework of unbounded operators in Hilbert spaces as a rigorous base for quantum mechanics. We then discuss Schrödinger operators (H = - ∆ + V) in this framework and analyze existence of solutions to the time-dependent Schrödinger equation, symmetries, time-dependent scattering theory, exponential localization of bound states, stability of the essential spectrum, potentially also perturbation theory and its convergence. Previous knowledge of quantum mechanics may be helpful, but is not required to follow the present course; we take an axiomatic approach here, starting from the abstractly stated framework of quantum mechanics.

Exam: The next appello is planned for the 21/1/2025. ONLY if you have NOT received the doodle link (in particular students that have not been attending the lectures this year): please write me an e-mail as soon as possible to agree on date and time of the oral exam.

Lecture notes part I: see 2017 course in Copenhagen further down on this page. List of topics:

  • Thursday, 3 October: review of the formalism of quantum mechanics, Banach spaces, Lp-spaces, bounded operators, Hilbert spaces, Fréchet-Riesz representation theorem, closed graph theorem, densely defined operators
  • Friday, 4 October: resolvent, spectrum, analytic functions, operator-valued analytic functions, Neumann series, spectrum of bounded operators
    • Exercises: Sheet 1 to be handed in Friday, October 11 before 10:30
  • Thursday, 10 October: adjoint operator, symmetric and self-adjoint operators, kernel, Hellinger-Töplitz theorem, Schwartz space, generalized theorem of the bounded inverse, basic criterion for self-adjointness
  • Friday, 11 October: solutions of the first exercise sheet
  • Thursday, 17 October: Kato-Rellich theorem, symmetry and self-adjointness of generator of the time evolution, self-adjoint extensions, Fourier transform
  • Friday, 18 October: Sobolev spaces, multiplication operators, Weyl criterion, Laplacian, Schrödinger operators
    • Exercises: Sheet 2 to be handed in Wednesday, October 23 before 20:59
  • Thursday, 24 October: Strongly Continuous Unitary Groups
  • Friday, 25 October: Symmetries, Quantum Noether theorem
  • Thursday, 31 October: Introduction to Scattering Theory
    • Exercises: Sheet 3 to be handed in Wednesday, November 13 before 20:59
  • Friday, 1 November: no lecture
  • Thursday, 7 November: no lecture
  • Friday, 8 November: no lecture
  • Thursday, 14 November: Scattering Theory: Dyson series, Asymptotic Completeness
  • Friday, 15 November: Scattering Theory: Lippmann-Schwinger Equation

New lecture notes part II:

Literature:

  • Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version
  • Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version
  • Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.
  • Elliott H. Lieb, Michael Loss: Analysis. Errata
  • Gerald B. Folland: A Course in Abstract Harmonic Analysis
  • Markus Haase: Lectures on Functional Calculus. Online version

An introduction to quantum mechanics was given in my course Fisica Matematica 3 (Meccanica Quantistica), Milan, Spring 2024. You can also read up on quantum mechanics from the physicist point of view in the books listed below.

  • Gordon Baym: Lectures on Quantum Mechanics
  • Leslie E. Ballentine: Quantum Mechanics, A Modern Development
  • Steven Weinberg: Lectures on Quantum Mechanics

Academic Year 2023-2024

Fisica Matematica 3 (Meccanica Quantistica), Milan, Spring 2024

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

  • Lunedì, 15 aprile: esperimenti del tipo Stern-Gerlach

    physicsworld: How the Stern–Gerlach experiment made physicists believe in quantum mechanics

  • Martedì, 16 aprile: descrizione di spin nello spazio vettoriale complesso di dimensione 2
  • Mercoledì, 17 aprile: base ortonormale per la direzione y dello spin
  • Giovedì, 18 aprile: osservabili (in)comptabili, relazione di indeterminazione, evoluzione temporale

    esercizi

  • Lunedì, 22 aprile: soluzione degli esercizi, osservabili continui (come la posizione)
  • Martedì, 23 aprile: operatori limitati su spazi di Hilbert in dimensione finita e infinita

    esercizi

  • Lunedì, 6 maggio: soluzione degli esercizi + operatore aggiunto
  • Martedì, 7 maggio: operatori illimitati
  • Mercoledì, 8 maggio: teoria spettrale

    esercizi, esempio esponenziale

  • Giovedì, 9 maggio: NO LEZIONE
  • Lunedì, 13 maggio: soluzione degli esercizi
  • Martedì, 14 maggio: simmetrie
  • Mercoledì, 15 maggio: rappresentazioni del gruppo di Galileo
  • Giovedì, 16 maggio: l'equazione di Schrödinger della particella libera

    esercizi con soluzione

  • Lunedì, 20 maggio: il gruppo di Galileo come gruppo di Lie
  • Martedì, 21 maggio: l'oscillatore armonico
  • Mercoledì, 22 maggio: l'atomo di idrogeno - aspetti qualitativi
  • Giovedì, 23 maggio: due lezioni, 13:30 - 15:30 & 15:30 - 17:30: spettro dell'atomo di idrogeno

    esercizi

  • Lunedì, 27 maggio: NO LEZIONE
  • Martedì, 28 maggio: NO LEZIONE
  • Mercoledì, 29 maggio: NO LEZIONE
  • Giovedì, 30 maggio: NO LEZIONE
  • Lunedì, 3 giugno: soluzione degli esercizi
  • Martedì, 4 giugno: NO LEZIONE
  • Mercoledì, 5 giugno: spettro dell'atomo di idrogeno
  • Giovedì, 6 giugno: autovettori angolari dell'atomo di idrogeno

    esercizi; soluzione in Balletine, Capitolo 10.1 "Spherical Potential Well"

  • Lunedì, 10 giugno: NO LEZIONE
  • Martedì, 11 giugno: disuguaglianza di Bell

Libri consigliati:

  • Bergfinnur Duurhus, Jan Philip Solovej: Mathematical Physics, lecture notes (inglese)
    versione pdf su https://noter.math.ku.dk/matematik.htm
  • Gerald Teschl: Mathematical Methods in Quantum Mechanics, With Applications to Schrödinger Operators (inglese)
    versione pdf sulla pagina web Prof. Teschl
  • Emmanuel Kowalski: Spectral theory in Hilbert spaces (Lecture Notes ETH Zürich, FS 09)
    appunti sulla pagina web Prof. Kowalski
  • David H. McIntyre: Quantum Mechanics, A Paradigms Approach (inglese)
  • Jun John Sakurai: Meccanica Quantistica Moderna, ristampa ampliata (italiano)
  • Leslie E. Ballentine: Quantum Mechanics, A Modern Development (inglese)
  • Brian C. Hall: Quantum Theory for Mathematicians, Springer Graduate Texts in Mathematics (inglese)
  • Michel Talagrand: What is a Quantum Field Theory? A First Introduction for Mathematicians (inglese)

Modalità d'esame: (Applicabile solo alla parte di meccanica quantistica, gestita independentemente dalla meccanica statistica.)

Per gli appelli di giugno e luglio, la prova scritta viene effetuata in modalità continua durante il semestre, tramite consegna settimanale degli esercizi. In questo caso, il voto per la prova scritta è solo "amesso alla prova orale" (> 50% dei punti degli esercizi, media per tutto il semestre) e "non ammesso alla prova orale".

Per gli appelli dopo luglio, ci sarà un esame scritto in presenza (modalità standard).

Le vostre soluzioni degli esercizi sono da consegnare al mio dottorando Diwakar Naidu, tramite mail diwakar.naidu@unimi.it o al ufficio 1010. La scadenza viene indicata con la pubblicazione degli esercizi sulla presente pagina web. Siete benvenuti di lavorare in gruppi e discutere gli esercizi, ma dovete essere in grado di spiegare le vostre soluzioni alla lavagna (quindi non copiate senza capire quello che scrivete).

Le date degli esami orali devono essere concordate individualmente con il docente in prossimità del relativo appello. I dettagli sulla procedura di iscrizione saranno pubblicati qui prima dell'appello.

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. 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, and stability of the essential spectrum.

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. Write me an email 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
  • 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
  • Wednesday, 6 December: no lecture
  • Friday, 8 December: no lecture
  • Wednesday, 13 December: spectral theorem
  • Friday, 15 December: spectral subspaces (1h) - exercise solutions (1h)
  • 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

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