We study a closed unbounded self-adjoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECs). We build a suitable MAEC for such a structure, prove it is ℵ0-categorical and ℵ0-stable up to a system of perturbations. We give an explicit continuous axiomatization for the class. We also characterize non-splitting and show it has the same properties as non-forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.