I have been studying open quantum systems for some time now. I have learnt about something known as spectral density that confers information about the physical structure and are found in the definition of the correlation functions. Now, the definition of spectral density is:
$$J(ω) = \sum_i ω^2_i λ^2_i δ(ω − ω_i)$$
where $ω_i$ are frequencies of the harmonic-oscillator bath modes and $λ_i$ are dimensionless couplings to the respective modes.
However, sometimes I have seen the spectral density defined as
where gamma is the reorganization energy and lambda is the cutoff frequency. I hear this is called Ohmic spectral density with Lorentz-Drude cutoff function, correct? How is this related to the formal definition of spectral density?
Also, I have seen that the spectral density is also written as
I hear this is also called Ohmic spectral density. My main question is, how is the second equation and third equation, both called Ohmic density, equal to each other or related to each other?