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Lecturer: Alessandro Michelangeli
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
office hours:
Tue, 15:00-16:00, office A-724
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.
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 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.
- 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 ℂ-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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Estimates on
eigenvalues. Lieb-Thirring inequalities.
Lieb-Thirring inequalities: statement and
meaning. Semiclassical
heuristics. Kinetic energy inequality. Birman-Schwinger principle.
Proof of Lieb-Thirring.
- 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).
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 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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