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Mathematical Quantum Mechanics

Course Type: 
PhD Course
Academic Year: 
Nov 10 - Feb 29
60 h

Web Page:

venue and schedule: room A-136, Mo 9:15-11:00 + Tue 11:15-13:00
start: 10 November 2015
end: 8 March 2016
duration: 60 hours (3 cycles); partial credits are possible

Synopsis: This course discusses the main mathematical problems that constitute the rigorous formalisation and treatment of a quantum mechanical model: the self-adjointness of the Hamiltonian, its stability, the spectral analysis, and the long-time behaviour of the dynamics (scattering properties). A number of operator-theoretic and functional-analytic tools will be introduced or reviewed, and their application to the rigorous study of such issues will be discussed. While more emphasis will be given on the self-adjointness and the stability problems, with reference also to some topical research lines in modern math-phys, spectral theory will be set up so as to be further developed in Dr De Oliveira's course, the study of the quantum many-body dynamics will be the object of Prof. Pickl's course, and scattering theory will be among the contents of Prof. Yajima's course, all scheduled after this.


  1. Preliminaries, general settings, main mathematical problems in QM.
    (1) First principles (a pragmatic survey). Axiomatics of Quantum Mechanics: finitely many vs infinite degrees of freedom. State and observables. Dynamics. Unitary evolution and the Schrödinger equation. Quantisation. Schrödinger's representation. (2) The quantum particle. States: spatial sector times spin. Typical one-body observables. Probabilistic interpretation: wave-function as probability density. Typical one-particle Hamiltonians: without spin (harmonic oscillator, hydrogenic atoms, semi-relativistic particle, ...) and with spin (the Zeeman effect). (3) Multi-particle formalism. Spin and Statistics. Tensor products of Hilbert spaces. Typical many-body Hamiltonians. (4) Four mathematical problems in QM
    (the `four-S prorgramme´): Self-adjointness of the Hamiltonian, Spectral analysis, Stability, Scattering theory.
  2. Self-adjoint operators on Hilbert space.
    Role of self-adjointness in QM. Paradoxes. Emergence of unboundedness in QM. Domain issues. Hamel basis. Hellinger-Toeplitz theorem. Graph of an operator. Closable and closed operators. Operator closure.  Algebraic properties. Core of an operator and of a closed operator. Adjoint of a densely defined operator. Multiplication operators. Examples of construction of the adjoint for differential operators on intervals. Algebraic properties of the adjoint. Relation between adjoint, closability, invertibility. Resolvent and spectrum of (possibly unbounded) closed operators. Empty spectrum or full -plane spectrum. Spectrum of bdd operators is non-empty and compact. Spectrum of multiplication operator. Essential range. Symmetric operators (not necess. densely defined). Semi-boundedness, positivity. Densely defined symmetric operators. Deficiency indices and their constance on each complex half-plane. Self-adjoint and essentially self-adjoint operators. Basic criteria of (essential) self-adjointness. von Neumann's formula. Spectrum of self-adjoint operators. Weyl's criterion and application to Schrödinger operators.
  3. Spectral theory.
    Spectral measures on Hilbert space (aka projection-valued measure). Characterisation of a pvm in terms of the associated scalar measures. Resolution of the identity and pvm. Support of a spectral measure. Spectral integrals of simple function, of bounded measurable functions, of unbounded measurable and a.e.-finite functions. Existence and main properties. Spectral theorem for bounded and for unbounded self-adjoint operators -- pvm form. Functional calculus. Paradigmatic examples of functional calculus. Main properties of functional calculus. Algebraic properties. Bounded case (functional calculus as a continuous *-homomorphism) and general case. Positivity, self-adjointness, square root via functional calculus. Characterisation of spectrum and resolvent via functional calculus. Stone's formulas. Spectral resolution and QM-interpretation (link to the axioms). Applications of the functional calculus: handy manipulation of functions of an operator. Commutativity in terms of spectral measures. Nelson's example. The Riesz projection. Estimating eigenspaces. Temple's inequality. Cyclic vectors and simple spectrum. Spectral basis. Spectral theorem in multiplication operator form. Spectral decomposition: point, continuous, absolutely continous, singular, singular continuous spectrum. Reduction to spectral subspaces. Examples. Wonderland theorem. Essential and discrete spectrum. Singular Weyl sequence. Relatively compact perturbations. Examples. Spectral theory for compact self-adjoint operators.
  4. Quantum dynamics.
    One-parameter strongly continuous unitary groups. Infinitesimal generator. Stone's theorem. Cores and Nelson's criterion. Translation group. Dilation group. Bounded infinitesimal generators.
    Lie product formula. Trotter product formula for the group and the semi-group of A+B. The case A+B self-adjoint and essentially self-adjoint. Differential equation on Hilbert spaces (Schrödinger, heat, wave equation) and their global well-posedness. Schrödinger unitary evolution with initial datum in the domain or outside the domain of the Hamiltonian. Regularisation effect of the heat equation at later times. Differential operator in d dimensions with constant coefficients. Minimal and maximal realisation, Fourier realisation. Convolution structure. Kernel of the contraction semi-group of the free negative Laplacian. Kernel of the free Schrödinger propagator. Green's function of the Laplacian. Lp-Lq interpolation. Riesz-Thorin interpolation theorem. L1-Linf e Lp-Lq dispersive estimates. Large times asymptotics of the free evolution. Finite speed of propagation. Strichartz estimates. MDFM formula. Smoothing. Higher regularity for the free Schrödinger equation.
  5. Methods for self-adjointness.
    Weyl's limit point-limit circle alternative. Relatively bounded perturbations. Kato smallness. Kato-Rellich theorem. Self-adjointness of Schrödinger operators via perturbation methods. Sobolev embedding. Controlling local singularities
    |x|^{-λ}. Kato and Hardy inequalities. Hardy-Littlewood-Sobolev inequality. Self-adjointness of magnetic Schrödinger operators. Diamagnetic inequality and Leinfelder-Simander theorems. Analytic vectors and free quantum fields.
  6. Quadratic (energy) forms.
    Quadratic forms and self-adjoint operators. Semi-bounded forms. Order relations. The Friedrichs extension. The Krein-von Neumann extension. Minimal and maximal Laplacian. Perturbation of forms and form sums. The Rollnik class and the KLMN theorem.
  7. Variational principle for Schrödinger operators.
    Domination of kinetic energy. Minimising sequences, compactness, weak convergence, lower semi-continuity. Existence of the ground state. Rellich-Kondrashev theorem and compact embedding. Excited states. Properties of eigenfunctions, regularity, and exponential decay. Harnack inequality. Min-max principle.
  8. Estimates on eigenvalues. Lieb-Thirring inequalities.
    Lieb-Thirring inequalities: statement and meaning. Semiclassical heuristics. Kinetic energy inequality. Birman-Schwinger principle. Proof of Lieb-Thirring.
  9. Stability of matter.
    Stability of hydrogenoid atoms. Stability of first and second kind. Three ingredients: Coulomb singularities, electrostatic screening, Pauli principle. Electrostatics and Baxter's inequality. Newton's theorem. Stability of molecular Hamiltonians (non-relativistic matter). Stability of matter via Thomas-Fermi theory. Sketch of stability of relativistic matter.

Pre-requisites: Physically: a first (undergraduate-like) exposition to the general framework of Quantum Mechanics would be useful to place the maths of the course into its physical context, but is not strictly needed, for the physics background/motivation will be discussed along the way. Mathematically: some very basic knowledge of functional analysis will be given for granted (only basic facts on Hilbert spaces, L^p spaces, distributions, Fourier transform): all the needed tools will be developed in class. The 20h course "Conceptual and Mathematical Foundations of Quantum Mechanics" that precedes this course is another excellent opportunity to get exposed to some of the needed pre-requisites. The course is also designed to intersect with a few talks on the subject, scheduled within the Analysis, Math-Phys, and Quantum series.

Exam: by one of the following procedures:

  • a public seminar where to discuss a research paper or other material related with the course, previously decided together with the instructor (intermediate discussions with the instructor are recommended before delivering the seminar)
  • a short essay (~10 pages) on themes previously agreed with the instructor
  • an oral examination
  • a 90' written test
  • a take-home exam (exercises to solve at home and to present to the instructor).

The examination panel will be formed by: L. Dabrowski, G. Dell'Antonio, G. De Oliveira, A. Michelangeli (SISSA), P. Pickl (LMU Munich), and K. Yajima (Tokyo).


Amrein, "Hilbert Space Methods in Quantum Mechanics", EPFL Press (2009)
Dell'Antonio, "Lectures on the Mathematics of Quantum Mechanics I", Springer (2015)

De Oliveira, "Intermediate Spectral Theory and Quantum Dynamics", Birkhäuser (2009)
Grubb, "Distributions and Operators", Springer (2009)
Lieb, Loss, "Analysis, Second Edition", AMS (2001)
Lieb, Seiringer, "The Stability of Matter in Quantum Mechanics", Cambridge (2010)
Reed and Simon, "Methods of Modern Mathematical Physics" vol I-IV, AP (1972-1980)
Schmüdgen, "Unbounded Self-adjoint Operators on Hilbert Space", Springer (2012)
Strocchi, "An Introdution to the Mathematical Structure of Quantum Mechanics", World Scientific (2008)
Teschl, "Mathematical Methods in Quantum Mechanics", AMS (2009)


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