= Solution
Write the membrane height above its mean plane as a sum of <Fourier modes>, $h(\mathbf r)=\sum_{\mathbf q}h_{\mathbf q}e^{i\mathbf q\cdot\mathbf r}$. The supplied average is a thermal mean-squared modal height; for complex Fourier coefficients it is understood as $\langle|h_{\mathbf q}|^2\rangle$, with $h_{-\mathbf q}=h_{\mathbf q}^*$ for a real height field. The terms are:
* $k_BT$ is the thermal energy, with <Boltzmann constant> $k_B$ and absolute <temperature> $T$.
* $A$ is the projected membrane <area> used in the Fourier normalization.
* $q=|\mathbf q|$ is the in-plane <wavenumber>; the corresponding <wavelength> is $2\pi/q$.
* $\sigma$ is the <membrane tension>, in $\mathrm{N\,m^{-1}}=\mathrm{J\,m^{-2}}$.
* $\kappa$ is the <membrane bending modulus>, in joules.
In the small-slope approximation, <Helfrich membrane energy> contains
$$
F=\frac12\int_A[\sigma|\nabla h|^2+\kappa(\nabla^2h)^2],d^2r
=\frac A2\sum_{\mathbf q}(\sigma q^2+\kappa q^4)|h_{\mathbf q}|^2.
$$
The <equipartition theorem>, with the independent real modes counted consistently, gives the stated <membrane fluctuation spectrum under tension>. Its units are length squared: $\sigma q^2$ and $\kappa q^4$ both have units $\mathrm{J\,m^{-4}}$, so $k_BT/[A(\sigma q^2+\kappa q^4)]$ has units $\mathrm{m^2}$.
At small length scales, $q$ is large and the bending term dominates:
$$
\boxed{\langle|h_{\mathbf q}|^2\rangle\simeq\frac{k_BT}{A\kappa q^4}
\quad\text{for }q\gg\sqrt{\sigma/\kappa}.}
$$
The fluctuation <amplitude> is controlled mainly by the <membrane bending modulus> and thermal energy, rather than tension. The crossover occurs when $\sigma q_c^2=\kappa q_c^4$, hence
$$
\boxed{q_c=\sqrt{\frac\sigma\kappa},\qquad
\ell_c\equiv q_c^{-1}=\sqrt{\frac\kappa\sigma},\qquad
\lambda_c=2\pi\sqrt{\frac\kappa\sigma}.}
$$
The bending-dominated regime extends up to the <membrane elastic length> when scale means $1/q$, or up to $\lambda_c$ when it means <wavelength>. At still larger scales tension gives a $q^{-2}$ spectrum. At the opposite end, the continuum formula requires <wavelengths> larger than the microscopic membrane thickness or molecular cutoff. Without numerical $\kappa$ and $\sigma$, no numerical crossover scale is determined.
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