# Mathematical Quantum Theory

**SUMMARY OF LECTURES**

**Lecture 1. **Introduction to quantum mechanics. The Schroedinger equation. Comparison with classical mechanics. Functional spaces: C^k and L^p spaces, Hilbert spaces.

**Lecture 2. **The free Schroedinger equation. The Fourier transform in L^1. Schwartz functions. Properties of the Schwartz space S: notions of convergence, metric. Completeness. The Fourier transform on S.

**Lecture 3. **Properties of the Fourier transform on S. Example: the Fourier transform of the Gaussian. Solution of the free Schroedinger equation for initial data in S.

**Lecture 4. **Decay of the wave packet. Pseudodifferential operators. Example: translations and the free propagator. Solution of the heat equation. Convolutions. Comparison between heat, wave and Schroedinger equations.

**Lecture 5. **Tempered distributions. Weak and weak* convergence. The adjoint map. Distributional Fourier transform and distributional derivative. Derivative of the delta and of the theta distributions.

**Lecture 6. **Density of S in S'. Formulation of the free Schroedinger equation on tempered distributions. Existence and uniqueness of the solution. Asymptotic behavior of the solution of the free Schroedinger equation.

**Lecture 7. **Interpretation of minus i times gradient as momentum operator. The Schroedinger equation with potential: introduction. Scalar product, Hilbert spaces. Bessel and Cauchy-Schwarz inequalities. Orthonormal bases. Isometry between separable Hilbert spaces and square summable sequences.

**Lecture 8. **Characterization of ONB of Hilbert spaces. Orthogonal complement. Splitting of Hilbert spaces in orthogonal subspaces. Bounded linear operators: definition, completeness of the space of bounded linear operators. Equivalence of boundedness of linear operators and continuity. Bounded extension of densely defined bounded linear operators.

**Lecture 9.** Completeness of the space of bounded linear operators implies completeness of the target space. The Fourier transform on L2. Unitary operators. The propagator of the free Schroedinger equation on L2. Notions of convergence for sequences of operators: norm, strong, weak. L2 solutions of the Schroedinger equation. Sobolev spaces.

**Lecture 10. **Strongly continuous 1-parameter group of unitary operators. Generator of a unitary group. Properties of the generator. Example: translations, on R and on a bounded interval. Domain of the generator. Sobolev lemma.

**Lecture 11. **Proof of the Sobolev lemma. Riesz theorem. Self-duality of Hilbert spaces. Hilbert space adjoint of a bounded linear operator. Properties of the adjoint. Example: the shift operator and its adjoint. The adjoint of a unitary operator.

**Lecture 12. **Bounded selfadjoint operators. Every bounded selfadjoint operator generates a unitary group. Unbounded operators. Adjoint operator. Unbounded selfadjoint operators. Example: translations on a bounded interval. Selfadjoint operators as generators of unitary groups (statement of the theorem). Graph of a linear operator.

**Lecture 13. **Closure of linear operators. The adjoint of a densely defined operator is closed. Extension of a symmetric operator via its adjoint. Properties of adjoints.

**Lecture 14. **Symmetric operators. Self-adjointness of multiplication operators. Self-adjointness of the Laplacian. Essential self-adjointness.

**Lecture 15. **The generator of translations is not essentially selfadjoint on a too small domain. Criteria for selfadjointness and essential selfadjointness. Self-adjoint extension of symmetric operators, completion of the domain.

**Lecture 16. **Quadratic form associated to a nonnegative operator. Form domain. Friedrichs extension. Reconstruction of nonnegative operators from the quadratic form: closable forms.

**Lecture 17. **Reconstruction of selfadjoint operators from their quadratic form. Resolvent set, resolvent, spectrum of a linear operator. Decomposition of the spectrum of closed operators. Properties of the resolvent and of the spectrum. The spectrum of selfadjoint operators is real.

**Lecture 18. **Symmetric operators with real spectrum are selfadjoint. Weyl criterion. Spectrum of the inverse operator. The spectral theorem: motivations, observables in quantum mechanics, solution of the Schrodinger equation.

**Lecture 19. **Projection valued measures. Properties. Functional calculus for simple functions. Extension to bounded Borel measurable functions.

**Lecture 20. **Functional calculus for unbounded Borel measurable functions. Properties.

**Lecture 21. **Construction of projection-valued measures. Borel transform, Herglotz functions. Characterization of Herglotz functions.

**Lecture 22. **Proof of the Herglotz theorem. The quadratic form of the resolvent defines a Herglotz function.

**Lecture 23. **Construction of the projection valued measure associated to a self-adjoint operator. Proof of the spectral theorem.

**Lecture 24. **The spectrum coincides with the support of the projection-valued measure. Self-adjoint operators are unitarily equivalent to multiplication operators.

**Lecture 25. **Absolutely continuous, singular continuous and pure point spectrum. Existence and uniqueness of the solution of the Schroedinger equation for general selfadjoint Hamiltonian.

**Lecture 26. **Dynamical properties of the absolutely continuous, singular continuous and pure point spectral subspaces. Wiener theorem. Compact operators. Dispersion at infinity, recurrent dynamics and localized states.

**Lecture 27. **Dynamical characterization of the spectral subspaces: RAGE theorem. Perturbations of selfadjoint operators: Kato-Rellich theorem.

**Lecture 28. **Proof of Kato-Rellich theorem. Application: hydrogenic atom. Coulomb uncertainty principle. Stability of the hydrogenic atom.