Let us first review some basic random graph models. The most popular is perhaps the Erdös-Rényi model, denoted by G(n, p) for some n ∈ N and p ∈ [0, 1]. In this model, there is a set of vertices V (with n = |V |), and the edge set over V is determined randomly by ( n 2 ) biased coin tosses. Initially, there are no edges: E := ∅. Then, for each pair of vertices u, v ∈ V , independently flip a co...

2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Lebesgue-Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . ...

This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are essentials in Quantum Mechanics: unbounded operators, multiplication operators, self-adjointness, spectrum, functional calculus, spectral measures and von Neumann’s Spectral Theorem. 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . ...

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