  We introduce, in the dual Macaev ideal of compact operators of a Hilbert
space, the spectral weight $\rho(L)$ of a positive, self-adjoint operator $L$
having discrete spectrum away from zero. We provide criteria for its
measurability and unitarity of its Dixmier traces ($\rho(L)$ is then called
spectral density) in terms of the growth of the spectral multiplicities of $L$
or in terms of the asymptotic continuity of the eigenvalue counting function
$N_L$. Existence of meromorphic extensions and residues of the $\zeta$-function
$\zeta_L$ of a spectral density are provided under summability conditions on
spectral multiplicities. The hypertrace property of the states
$\Omega_L(\cdot)={\rm Tr\,}_\omega (\cdot\rho(L))$ on the norm closure of the
Lipschitz algebra $\mathcal{A}_L$ follows if the relative multiplicities of $L$
vanish faster than its spectral gaps or if $N_L$ is asymptotically regular.
