When can you neglect terms like $a_n a_{n+1}$ or $a_n^\dagger a_{n+1}^\dagger$? [duplicate]

Question

I am looking for a detailed explanation of when and why we can neglect terms like $a_n a_{n+1}$ or $a_n^\dagger a_{n+1}^\dagger$ in Hamiltonians like $$ H = \omega \sum_n a^\dagger_n a_n + \lambda\sum_{n} \left(a_n a_{n+1} + a_n^\dagger a_{n+1}^\dagger + a^\dagger_n a_{n+1} +a^\dagger_{n+1}a_n \right) \approx \omega \sum_n a^\dagger_n a_n + \lambda\sum_{n} \left( a^\dagger_n a_{n+1} +a^\dagger_{n+1}a_n \right).$$ I've seen this arise in the context of quantum circuits and other models where one encounters interactions of the form $x_n x_{n+1}$ , $p_n p_{n+1}$, $x_n p_{n+1}$, etc. The typical explanation given is that "$\omega \gg \lambda$" or that "in the rotating wave approximation, terms like $a_n a_{n+1}$ oscillate fast and can be ignored".

In the rotating frame, I see that transformations like $$H(t) = \exp(-it\omega\sum_n a^\dagger_n a_n)H\exp(it\omega\sum_n a^\dagger_n a_n)$$ leave a fast oscillating factor in front of terms like $a_n a_{n+1}$ but why/when can it be disregarded?

I don't see how $\omega \gg \lambda$ is relevant for the approximation.

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Edit 1, based on the comments:

1. I will stress that I am looking for a way of justifying neglecting $\lambda \sum_n (a_n a_{n+1} + a_n^\dagger a_{n+1}^\dagger)$ but NOT the hopping term $\lambda \sum_n (a_n^\dagger a_{n+1} + a_n a_{n+1}^\dagger)$. I suspect this will likely involve talking about the spectrum/eigenvectors of $H$ in a particular limit.
2. I have for instance seen this arise in the context of quantum electrical circuits. For instance, one can approximate a chain of capacitively/inductively coupled harmonic LC oscillators to a hopping chain. That is, I can show that the Hamiltonian of a chain of capacitively/inductively coupled harmonic LC oscillators is $$ H = \sum_n \alpha_n p_n^2 + \beta_n x_n^2 + \lambda_n x_n x_{n+1}.$$ Note that the coupling term $x_n x_{n+1} = (a^\dagger_n + a_n)(a^\dagger_{n+1} + a_{n+1})$ yields exactly my first Hamiltonian. However, following reference "Bloch oscillations and Wannier-Stark ladder in the coupled LC circuits (Bahmani 2020)" equation (12), one claims that if $\omega \gg \lambda$ then $$H \approx \sum_n \omega_n a^\dagger_n a_n + \lambda_n\sum_{n} \left( a^\dagger_n a_{n+1} +a^\dagger_{n+1}a_n \right),$$ if I understand the ref correctly.