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by Gerald Teschl (mat.univie.ac.at)

TOC

Preliminaries
A First look at Banach and Hilbert spaces
1. Warm up: Metric and topological spaces
2. The Banach space of continuous functions
3. The geometry of Hilbert spaces
4. Completeness
5. Bounded operators
6. Lebesgue Lp spaces
7. Appendix: The uniform boundedness principle

Part 1: Mathematical Foundations of Quantum Mechanics
# Hilbert spaces

1. Hilbert spaces
2. Orthonormal base
3. The projection theorem and the Riesz lemma
4. Orthogonal sums and tensor products
5. The C* algebra of bounded linear operators
6. Weak and strong convergence
7. Appendix: The Stone-Weierstra theorem

# Self-adjointness and spectrum

1. Some quantum mechanics
2. Self-adjoint operators
3. Quadratic forms and the Friedrichs extension
4. Resolvents and spectra
5. Orthogonal sums of operators
6. Self-adjoint extensions
7. Appendix: Absolutely continuous functions

# The spectral theorem

1. The spectral theorem
2. More on Borel measures
3. Spectral types
4. Appendix: The Herglotz theorem

# Applications of the spectral theorem

1. Integral formulas
2. Commuting operators
3. The min-max theorem
4. Estimating eigenspaces
5. The spectra of tensor products

# Quantum dynamics

1. The time evolution and Stone’s theorem
2. The RAGE theorem
3. The Trotter product formula

# Perturbation theory for self-adjoint operators

1. Relatively bounded operators and the Kato-Rellich theorem
2. More on compact operators
3. Hilbert-Schmidt and trace class operators
4. Relatively compact operators and Weyl’s theorem
5. Relatively form bounded operators and the KLMN theorem
6. Strong and norm resolvent convergence

Part 2: Schrödinger Operators
# The free Schrödinger operator

1. The Fourier transform
2. The free Schrödinger operator
3. The time evolution in the free case
4. The resolvent and Green’s function

# Algebraic methods

1. Position and momentum
2. Angular momentum
3. The harmonic oscillator
4. Abstract commutation

# One dimensional Schrödinger operators

1. Sturm-Liouville operators
2. Weyl’s limit circle, limit point alternative
3. Spectral transformations I
4. Inverse spectral theory
5. Absolutely continuous spectrum
6. Spectral transformations II
7. The spectra of one-dimensional Schrödinger operators

# One-particle Schrödinger operators

1. Self-adjointness and spectrum
2. The hydrogen atom
3. Angular momentum
4. The eigenvalues of the hydrogen atom
5. Nondegeneracy of the ground state

# Atomic Schrödinger operators

1. Self-adjointness
2. The HVZ theorem

# Scattering theory

1. Abstract theory
2. Incoming and outgoing states
3. Schrödinger operators with short range potentials

Part 3: Appendix
# Almost everything about Lebesgue integration

1. Borel measures in a nut shell
2. Extending a premeasure to a measure
3. Measurable functions
4. The Lebesgue integral
5. Product measures
6. Vague convergence of measures
7. Decomposition of measures
8. Derivatives of measures

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