set of eigenvalues of an observable may be continuous rather than discrete. An example is the position observable, . The spectrum of the position (in one dimension) is the entire real line and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics with an appropriate rigged Hilbert space. In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by The eigenfunctions of position are denoted by in Dirac notation, and are known as position eigenstates. Similar considerations apply to the eigenstates of the momentum operator, or indeed any other self-adjoint unbounded operator P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φy of distributions indexed by the elements of Ω, such that That is, φy are the eigenvectors of P. If the eigenvectors are normalized so that in the distribution sense, then for any test function ψ, where That is, as in the discrete case, there is a resolution of the identity where the operator-valued integral is again understood in the weak sense. If the spectrum of P has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum. The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well. Structural mechanics The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse I at time t = 0 can be written where m is the mass, ξ the deflection and k the spring constant. As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory, where EI is the bending stiffness of the beam, w the deflection, x the spatial coordinate and q(x) the load distribution. If a beam is loaded by a point force F at x