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Description of the course
The course consists of two parts. General results for self-adjoint operators will
be presented, with particular emphasis on Schrödinger operators, corresponding to
quantum Hamiltonians of physical interest. We will first describe various approaches
to the description of self-adjoint extensions of symmetric operators. Then we shall
deal with different classifications of the spectrum of self-adjoint operators, presenting
some results about stability under perturbations. The relation with the associated
quantum dynamics will be presented. This subject will be further explored in the
second part of the course, dealing with quantum scattering problems. We will focus
mainly on the one-particle case, presenting the basic aspects of the theory, in both the
time-independent and time-dependent formulations.
The course is organized as follows.
First part (W. Borrelli - 12 hours, approx.)
(1) Self-adjoint extensions of symmetric operators: von Neumann theory and resolvent
techniques. Quadratic form methods.
(2) Applications to Schrödinger operators.
(3) (Time permitting) Magnetic Schrödinger operators.
(4) Classification of the spectrum of self-ajoint operators and quantum dynamics.
(5) Perturbation theory for self-adjoint operators: stability of the essential spectrum
and existence of discrete eigenvalues. Min-max principles for eigenvalues
of Schrödinger operators.
Second part (D. Fermi - 13 hours, approx.)
(1) A review of operator ideals: compact, Hilbert-Schmidt and trace class operators.
(2) Non-relativistic one-body scattering theory: characterization of asymptotically
free states, existence and completeness of wave operators, time-dependent and
stationary methods (Kato-Birman theory), the role of the singular spectrum,
generalized eigenfunctions and the Lippmann-Schwinger equation (Agmon theory
and the Limiting Absorption Principle).
(3) Definition of the scattering operator and its connection to the physical cross
section. Perturbative expansions: Born and Dyson series.
At the end of the course the student shall be able to:
- analyze basic spectral properties of self-ajoint operators
- understand the mathematical foundations of scattering theory in non-relativistic quantum mechanics
- rigorously formulate basic mathematical problems related to the study of non-relativistic quantum dynamics
Prerequisites.
Basic knowledge of functional analysis and operator theory in Hilbert spaces, Schwartz distributions and Sobolev spaces, an introductory course on quantum mechanics.
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Inglese