Self-adjoint operators in Quantum Mechanics 21/22

• [5/5/2022] Axioms of Quantum Mechanics, the need for self-adjointness and unboundedness. (Refs: [T] Chapter 5, [GM] Appendix A, [M] Chapter 7);

• [23/5/2022] Basic definitions with examples (Unbounded, closed, closable operators). (Ref. [S] Section 1.1);

• [24/5/2022] Adjoint: definition, basic properties and examples. (Ref . [S] Chapter 1);

• [25/5/2022] Resolvent, spectrum and symmetric operators. (Ref. [S] Sections 2.2, Section 3.1).

• [26/5/2022] Properties of symmetric operators, deficiency indices, self-adjoint operators. von Neumann extension scheme and examples (Refs. [S] Section 3.2, [GM] Section 2.3)

• [27/5/2022] Spectral theorem, functional calculus and applications: well-posedness of Schrodinger, heat and wave equations.

• [30/5/2022] Relatively bounded perturbations of self-adjoint operators. Kato-Rellich and applications: atomic Hamiltonians and Dirac Operator (Ref. [S] Section 8.2, [GM] Section 4.1, [RS] Section X.2).

• [31/5/2022] Quadratic forms, Friedrichs extension, KLMN Theorem. Krein-Vishik-Birman extension scheme (Refs. [RS] Section VIII.6, X.3; [S] Section 10.7; [GM] Sections 2.7-2.9).

• [1/6/2022] Example of application of the KVB extension scheme.

References

[GM] M. Gallone, A. Michelangeli, Self-adjoint extension schemes and modern applications to quantum Hamiltonians, arXiv:2201.1025
[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