
Lecturer: Alessandro Michelangeli
venue and
schedule: room A136, Mo 9:1511:00 + Tue 11:1513:00
start: 10 November
2015
end: 8 March
2016
duration: 60 hours (3
cycles); partial credits are possible
office hours:
Tue, 15:0016:00, office A724
Synopsis: This
course discusses the main mathematical problems that constitute the
rigorous formalisation and treatment of a quantum mechanical model: the
selfadjointness of the Hamiltonian, its stability, the spectral
analysis, and the longtime behaviour of the dynamics (scattering
properties). A number of operatortheoretic and functionalanalytic
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 selfadjointness and the stability problems, with
reference also to some topical research lines in modern mathphys,
spectral theory will be set up so as to be further developed in Dr
De Oliveira's course, the study of the quantum manybody 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.
Topics:
 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 onebody observables. Probabilistic interpretation:
wavefunction as probability density. Typical oneparticle
Hamiltonians: without spin (harmonic oscillator, hydrogenic atoms,
semirelativistic particle, ...) and with spin (the Zeeman effect). (3)
Multiparticle formalism. Spin and Statistics. Tensor products of
Hilbert spaces. Typical manybody Hamiltonians. (4) Four mathematical
problems in QM (the `fourS
prorgramme´): Selfadjointness
of the Hamiltonian, Spectral
analysis, Stability, Scattering theory.
 Selfadjoint
operators on Hilbert space.
Role
of selfadjointness in QM. Paradoxes. Emergence of unboundedness in QM.
Domain issues. Hamel basis. HellingerToeplitz 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 ℂplane
spectrum. Spectrum of bdd operators is nonempty and compact. Spectrum
of multiplication operator. Essential range. Symmetric operators (not
necess. densely defined). Semiboundedness, positivity. Densely defined
symmetric operators. Deficiency indices and their constance on each
complex halfplane. Selfadjoint and essentially selfadjoint
operators. Basic criteria of (essential) selfadjointness. von
Neumann's formula. Spectrum of selfadjoint operators. Weyl's criterion
and application to Schrödinger operators.
 Spectral theory.
Spectral measures on Hilbert space (aka projectionvalued 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 selfadjoint
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, selfadjointness, square
root via functional calculus. Characterisation of spectrum and
resolvent via functional calculus. Stone's formulas. Spectral
resolution and QMinterpretation (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 selfadjoint operators.
 Quantum dynamics.
Oneparameter 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
semigroup of A+B. The case A+B selfadjoint and essentially
selfadjoint. Differential
equation on Hilbert spaces
(Schrödinger, heat, wave equation) and their
global wellposedness. 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 semigroup of the free negative Laplacian. Kernel of the
free Schrödinger
propagator. Green's function of the Laplacian. LpLq interpolation.
RieszThorin interpolation theorem. L1Linf e LpLq 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.
 Methods for
selfadjointness.
Weyl's limit pointlimit circle alternative. Relatively bounded
perturbations. Kato smallness. KatoRellich theorem. Selfadjointness
of Schrödinger operators via perturbation methods. Sobolev embedding.
Controlling local singularities x^{λ}.
Kato and Hardy inequalities. HardyLittlewoodSobolev inequality.
Selfadjointness of magnetic Schrödinger
operators. Diamagnetic inequality and LeinfelderSimander
theorems. Analytic vectors and free quantum fields.
 Quadratic (energy)
forms.
Quadratic forms and selfadjoint operators. Semibounded forms. Order
relations. The Friedrichs extension. The Kreinvon Neumann extension.
Minimal and maximal Laplacian. Perturbation of forms and form sums. The
Rollnik class and the KLMN theorem.
 Variational
principle for Schrödinger operators.
Domination of kinetic energy. Minimising sequences, compactness, weak
convergence, lower semicontinuity. Existence of the ground state.
RellichKondrashev theorem and compact embedding. Excited states.
Properties of eigenfunctions,
regularity, and exponential decay. Harnack inequality. Minmax
principle.
 Estimates on
eigenvalues. LiebThirring inequalities.
LiebThirring inequalities: statement and
meaning. Semiclassical
heuristics. Kinetic energy inequality. BirmanSchwinger principle.
Proof of LiebThirring.
 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
(nonrelativistic matter). Stability of matter via ThomasFermi theory.
Sketch of stability of relativistic matter.
Prerequisites: Physically:
a first (undergraduatelike) 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
prerequisites. The course is also designed to
intersect with a few talks on the subject, scheduled within the Analysis,
MathPhys, 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 takehome 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).
Literature:
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 IIV, AP (19721980)
Schmüdgen, "Unbounded Selfadjoint
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)

