Frequency filtering changes a thermal spectrum

Let ${\omega }_1,{\omega }_2>0$ be distinct angular frequencies measured in seconds${}^{-1}$, let $\theta >0$ be measured in seconds, and choose a pointwise channel transmissivity $\mathrm{\Gamma }:(0,\mathrm{\infty })\to [0,1]$ with $\mathrm{\Gamma }({\omega }_1)=1$ and $\mathrm{\Gamma }({\omega }_2)=0$. Define the numerical Planck occupation profile $n_{\theta }(\omega )=(\exp (\theta \omega )-1{)}^{-1}$ and its filtered profile $m_{\mathrm{\Gamma }}(\omega ):=\mathrm{\Gamma }(\omega )n_{\theta }(\omega )$. The proposition's modewise multiplier formula motivates this pointwise profile map; this example does not assert that one normalized Fock-state family realizes the full profile. Then $m_{\mathrm{\Gamma }}$ is positive at ${\omega }_1$ and zero at ${\omega }_2$. No single Planck occupation $n_{{\theta }^{\mathrm{\prime }}}(\omega )$ with finite ${\theta }^{\mathrm{\prime }}>0$ measured in seconds has both values. This is a constructed mathematical filter, not a claim about an actual black-hole potential.

Exponential channel thermality and greybody filtered occupations

Nonlogical channel motivation. Exponential channel thermality and greybody filtered occupations proves the loss-channel identity ${\mathfrak{n}}_{\mathrm{o}\mathrm{u}\mathrm{t}}(\omega )=\mathrm{\Gamma }(\omega )n_{\omega }$ for each fixed-frequency normalized signal vector in the stated annihilation-operator domain and a vacuum environment. The profile $m_{\mathrm{\Gamma }}$ here is only the numerical pointwise multiplier of $n_{\theta }$; no state-family realization is asserted.