It is my understanding that in quantum mechanics we use self-adjoint operators (that is an axiom of the theory). This operators can be either bounded or unbounded, being the latter the more general case.
Has the spectral theorem had been proven for the unbounded case? And if so, what references discuss this?
The original source of the spectral theorem for unbounded self-adjoint operators was John von Neumann, who was a student of Hilbert. His formulation and proof can be found in the English translation Mathematical Foundations of Quantum Mechanics.
von Neumann uses the suggestive notation$$ H=\int \lambda \, \mathrm{d}{E \left( \lambda \right)}$$ for his spectral integral. von Neumann discusses the difference between symmetric operators and self-adjoint operators; he formulates abstract boundary conditions and discusses how to obtain a self-adjoint operator from the adjoint of a symmetric operator by imposing such conditions.
von Neumann's treatment is the original source and readable.