To give a concrete example, a free particle can have any energy it likes - it's only bound states that have discrete spectra. Mathematically, this corresponds

To give a concrete example, a free particle can have any energy it likes - it's only bound states that have discrete spectra. Mathematically, this corresponds to solutions to a particular differential equation existing for particular values of energy (which appears as a constant in the equation). To use a simpler DE for an example: dx/dt = kx has solutions Ce^kt for all k, but a more complicated DE might only have solutions for some k.