 The rigged Hilbert space representation of these observables is about as close as one can get to Dirac’s elegant and extremely useful formal representation with the added feature of being placed within a mathematically rigorous framework. It should be noted, however, that there is a sense in which it is a proper generalization of Dirac’s framework. The rigging (based on the choice of a nuclear operator that determines the test function space) can result in different sets of generalized eigenvalues being associated with an operator. For example, the set of (generalized) eigenvalues for the momentum operator (in one dimension) corresponds to the real line, if the space of test functions is the set \(S\) of infinitely differentiable functions of \(x\) which together with all derivatives vanish faster than any inverse power of \(x\) as \(x\) goes to infinity, whereas its associated set of eigenvalues is the complex plane, if the space of test functions is the set \(D\) of infinitely differentiable functions with compact support (i.e., vanishing outside of a bounded region of the real line). If complex eigenvalues are not desired, then \(S\) would be a more appropriate choice than \(D\) – see Nagel (1989) for a brief discussion. But there are situations in which it is desirable for an operator to have complex eigenvalues. This is so, for example, when a system exhibits resonance scattering (a type of decay phenomenon), in which case one would like the Hamiltonian to have complex eigenvalues – see Böhm &amp; Gadella (1989). (Of course, it is impossible for a self-adjoint operator to have complex eigenvalues in a Hilbert space.)