Lecture 1 14.10

Subjects: A motivation for quantum mechanics, a very heuristic derivation of the Schödinger equation. What does mathematical Methods of Quantum Mechanics mean?

Lecture 2 16.10

Subjects: An introduction to weak derivatives and Sobolev spaces

Lecture 3 18.10

Subjects: Unbounded operators, Kernel, Range, closed operators, Graph Norm, resolvent set, spectrum, resolvent

Lecture 4 25.10

Subjects: Basic Properties of the spectrum, resolvent set and the resolvent (closedness, openness, analyticity, respectively), spectrum of multiplication operators and Laplace, closable operators, criterium for closability

Lecture 5 28.10

Subjects: The adjoint operator and basic properties, symmetric and self-adjoint operators.

**Typos: In the proof of Thm 3.2 part 3 in page 4 in the two limits there should be a instead of a y.**

Lecture 6 30.10

Subjects: Properties of symmetric operators, the basic criterion of self-adjointness, self-adjointness of the Laplacian, Kato-Rellich Theorem

Lecture 7 06.11

Subjects: Proof of Kato Rellich Theorem, Retour in Convergence Theorems, Hardy's inequality, self-adjointness of the Hamiltonian of the hydrogen atom

Lecture 8 08.11

Subjects: Essential self-adjointness, basic criterion and Kato-Rellich Theorem for essential self-adjointness, Solutions of Schrödinger Equation

Lecture 9 13.11

Subjects: Essential self-adjointness necessary condition for existence of dynamics, strongly continuous unitary groups, Existence and properties of dynamics in the bounded self-adjoint case.

Lecture 10 13.11

Subjects: Existence and properties of dynamics in the self-adjoint case

Lecture 11 20.11

Subjects: Generator of a strongly continuous unitary group, Stone's Theorem

Lecture 12 22.11

Subjects: Observables, expectations and variance of observables, examples, uncertainty principle, ground state energy

Lecture 13 27.11

Subjects: Properties of the ground state energy, existence and uniqueness of a ground state energy of the hydrogen atom

Lecture 14 29.11

Subjects: Weak convergence, Banach Alaoglou Theorem, Compact Operators, Cauchy's Theorem, and Cauchy's integralformula for the resolvent of an operator

Lecture 15 04.12

Subjects: Spectral radius and its properties, Decomposition of a self-adjoint operator

Lecture 16 06.12

Subjects: Decomposition of a self-adjoint operator, sequencial criterion for the spectrum

Lecture 17 11.12

Subjects: Discrete and Essential spectrum, ionization threshold, Weyl's criterium for the essential spectrum

Lecture 18 13.12

Subjects: Retour to compact operators, Applications of Weyl's criterion

Lectures 19-20 18.12-20.12

Subjects: Schrödinger Operators with Potential growing to infinity, IMS localization formula, HVZ Theorem

Lecture 21 08.01

Subjects: Further properties of the essential spectrum, ionization threshold, problem of ionizition energies, an introduction to exponential decay of eigenfunctions.

Lecture 22 10.01

Subjects: Proof of exponential decay of eigenfunctions to eigenvalues below the ionization threshold

Lecture 23 15.01

Subjects: Spectral Theorem for self-adjoint operators.

Lectures 24-25 17.01-22.01

Subjects: An Elementary Proof of minimax Principle in a special case, corollaries, discrete spectrum of hydrogen atom, Newton's Theorem

Lecture 26 24.01

Subjects: Zhislin's Theorem, spectral measure of a self-adjoint operator

Lecture 27 29.01

Subjects: Properies of the spectral measure and proof that its support is equal to the spectrum of the operator

Lecture 28 31.01

Subjects: Further properties of the spectral measure, proof of the minimax principle, pure point spectrum and continuous spectrum, RAGE Theorem

**Correction: The pure point spectrum of a self-adjoint operator is defined to be the set of its eigenvalues. In the file of lectures 29-30 corrects the mistake**

Lectures 29-30 05.02-07.02

Subjects: pure point spectrum and continuous spectrum, RAGE Theorem examples and proof