  The quantum theory of the damped harmonic oscillator has been a subject of
continual investigation since the 1930s. The obstacle to quantization created
by the dissipation of energy is usually dealt with by including a discrete set
of additional harmonic oscillators as a reservoir. But a discrete reservoir
cannot directly yield dynamics such as Ohmic damping (proportional to velocity)
of the oscillator of interest. By using a continuum of oscillators as a
reservoir, we canonically quantize the harmonic oscillator with Ohmic damping
and also with general damping behaviour. The dynamics of a damped oscillator is
determined by an arbitrary effective susceptibility that obeys Kramers-Kronig
relations. This approach offers an alternative description of nano-mechanical
oscillators and opto-mechanical systems.
