Self-adjoint operators in Quantum Mechanics 23/24

Content

• [4/10/2023] Axioms of Quantum Mechanics: the need for self-adjointness and unboundedness ([T, Chapter 5], [GM, Appendix A], [D, Chapter 3], [M, Chapter 7, Section 11.1, Section 11.4, Section 13.1.1, Section 13.4]).

• [6/10/2023] Unbounded operators on Hilbert spaces. Definitions and examples, Graph of an operator, extensions and restrictions. Sobolev spaces H^s. Closed and closable operators ([S, Section 1.1], [M, Section 5.1.1-5.1.2], [GM, Section 1.2], [T, Sections 4.1-4.2]). [Exercise Sheet #1 - updated version 11/10]

• [9/10/2023] Final remarks on closed/closable operators.  Adjoint of an operator: examples, properties. Resolvent and spectrum for a closed operator. ([GM, Sections 1.3, 1.7], [S, Sections 1.2, 1.3, 2.2]).

• [11/10/2023] Symmetric and Self-adjoint operators. Characterization of self-adjoint operators. von Neumann decomposition formula for D(T^*) and von Neumann formula for self-adjoint extensions. ([GM, Section 1.8], [S, Sections 3.1-3.2]) [Exercise Sheet #2 - updated version 16/10]

• [13/10/2023] Spectral theorem, spectral measures and spectral integrals. Functional calculus ([GM,Sections 1.10, 1.11], [S, Section 4, 5.1, 5.2])

• [16/10/2023] Stone's theorem. Well-posedness for Schroedinger equation and heat equation ([GM, Section A3], [S, Section 6.1, 6.2], [D, Section 5.7]) [Exercise Sheet #3]

• [20/10/2023] Classification of spectra of Self-adjoint operators. Weyl sequence and characterization of the essential spectrum. Kato-Rellich theorem ([S, Sections 9.1, 8.4, 8.2], [GM, Sections 1.13, 1.14], [T, Proposition 4.8]).

• [23/10/2023] Quadratic forms, basic definitions and properties. Representation theorem for symmetric, closed and semi-bounded quadratic forms. Order relations for Self-adjoint operators. Perturbations of forms, KLMN Theorem. ([GM, Section 1.15], [S, Sections 10.1, 10.2, 10.3, 10.7]).

• [26/10/2023] Minimal and maximal realizations of formal differential operators. Friedrichs extension. Generalities on self-adjoint extension schemes. recap on von Neumann scheme. KVB extension scheme: operators, quadratic forms and properties of the extensions. ([GM, Sections 1.4, 2.1, 2.3, 2.7]) [Exercise Sheet #4] [Exam-like test]

• [30/10/2023] Shur's lemma. Exponential decay for the Green function and the Fermi projector for gapped Hamiltonians on lattice. Example of self-adjoint laplacian with irregular domain.

Some interesting questions are collected in the [Q&A Sheet] that will be updated during the course.

References

[D] G.F. dell'Antonio, Lectures  on the mathematics of quantum mechanics, Springer
[GM] M. Gallone, A. Michelangeli, Self-adjoint extension schemes and modern applications to quantum Hamiltonians, Springer
[M] V. Moretti, Spectral Theory and Quantum Mechanics, Springer
[RS]  M. Reed, B. Simon, Methods of modern Mathematical Physics , Elsevier
[S] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer
[T] A. Teta, A Mathematical Primer on Quantum Mechanics, Springer