03/21/2013 - 15:00
03/21/2013 - 17:00
Speaker:
Jan Derezinski, (University of Warsaw)
Location:
BURN 1120
Abstract:
Mini-Course in Analysis/Mathematical Physics (March 13, 14, 20, 21, 27; 3-5 pm, 10 hours total)
The main purpose of the course is to develop general theory of perturbations of linear operators on Hilbert spaces, with the emphasis on Schrödinger operators. Many concrete examples will be described in detail. These examples illustrate a number of interesting points relevant for quantum mechanics and probability theory.
List of subjects that will be (partially) covered:
1) Reminder of basic spectral theory
- Unbounded operators
- Closed operators
- Spectrum
- Pseudoresolvents
- Unbounded operators on Hilbert spaces
- (Essential) self-adjointness
- Relative boundedness
- Scale of Hilbert spaces
- Closed and closable positive forms
- Relative form boundedness
- Friedrichs extensions
2) Reminder of basic harmonic analysis and its applications
- Young inequality
- Sobolev inequalities
- Application: self-adjointness of Schrödinger operators
3) Momentum and Laplacian in 1 dimension
- Momentum on half-line
- Momentum on an interval
- Laplacian on half-line
-Laplacian on an interval
4) Orthogonal polynomials
- Orthogonal polynomials in weighted L^2 spaces
- Self-adjointness of Sturm-Liouville operators
- Classical orthogonal polynomials as eigenvectors of certain
Sturm-Liouville operators
- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
5) Finite rank perturbations and their renormalization
- Aronszajn-Donoghue Hamiltonians
- Delta potentials
- Friedrichs Hamiltonians
- Bound states and resonances of Friedrichs Hamiltonians
- Exponential decay from a unitary dynamics
6) Potential 1/|x|²
- Hardy inequality
- Modified Bessel equation
- Bessel equation
- Operator -∂²+(m²-1/4)/|x|²
List of subjects that will be (partially) covered:
1) Reminder of basic spectral theory
- Unbounded operators
- Closed operators
- Spectrum
- Pseudoresolvents
- Unbounded operators on Hilbert spaces
- (Essential) self-adjointness
- Relative boundedness
- Scale of Hilbert spaces
- Closed and closable positive forms
- Relative form boundedness
- Friedrichs extensions
2) Reminder of basic harmonic analysis and its applications
- Young inequality
- Sobolev inequalities
- Application: self-adjointness of Schrödinger operators
3) Momentum and Laplacian in 1 dimension
- Momentum on half-line
- Momentum on an interval
- Laplacian on half-line
-Laplacian on an interval
4) Orthogonal polynomials
- Orthogonal polynomials in weighted L^2 spaces
- Self-adjointness of Sturm-Liouville operators
- Classical orthogonal polynomials as eigenvectors of certain
Sturm-Liouville operators
- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
5) Finite rank perturbations and their renormalization
- Aronszajn-Donoghue Hamiltonians
- Delta potentials
- Friedrichs Hamiltonians
- Bound states and resonances of Friedrichs Hamiltonians
- Exponential decay from a unitary dynamics
6) Potential 1/|x|²
- Hardy inequality
- Modified Bessel equation
- Bessel equation
- Operator -∂²+(m²-1/4)/|x|²