Editing Spectrum (functional analysis) (section)

==Example: Hydrogen atom==

The [[hydrogen atom]] provides an example of different types of the spectra.  The [[molecular Hamiltonian|hydrogen atom Hamiltonian operator]] <math>H=-\Delta-\frac{Z}{|x|}</math>, <math>Z > 0</math>, with domain <math>D(H) = H^1(\R^3)</math> has a discrete set of eigenvalues (the discrete spectrum <math>\sigma_{\mathrm{d}}(H)</math>, which in this case coincides with the point spectrum <math>\sigma_{\mathrm{p}}(H)</math> since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the [[Rydberg formula]].  Their corresponding [[eigenfunction]]s are called '''eigenstates''', or the [[bound state]]s. The result of the [[ionization]] process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by <math>\sigma_{\mathrm{cont}}(H)=[0,+\infty)</math> (it also coincides with the essential spectrum, <math>\sigma_{\mathrm{ess}}(H)=[0,+\infty)</math>).
{{Citation needed|date=August 2019}}{{Clarify|date=November 2023}}