Take the resolvent matrix element $G(z) = \langle v_0 | (z\mathbb{1} - A)^{-1} | v_0 \rangle$ and evaluate it just off the real axis at $z = \omega + i\eta$. In the eigenbasis it is a sum of simple poles,

$$G(\omega + i\eta) = \sum_k \frac{w_k}{\omega - \lambda_k + i\eta}, \qquad w_k = |\langle v_0 | \psi_k\rangle|^2,$$

and since $\operatorname{Im}\frac{1}{x + i\eta} = \frac{-\eta}{x^2+\eta^2}$, the spectral function is a sum of Lorentzians:

$$A(\omega) = -\tfrac{1}{\pi}\operatorname{Im} G(\omega+i\eta) = \sum_k w_k ,\frac{\eta/\pi}{(\omega-\lambda_k)^2 + \eta^2}.$$

A Lorentzian fit to a single peak reads off three numbers: its center is the eigenvalue $\lambda_k$ (the resonant frequency), its area is the overlap weight $w_k$ (an oscillator strength — how brightly $v_0$ couples to that mode), and its width is $\eta$ (an artificial broadening for a discrete matrix, or the mode’s physical inverse lifetime $\Gamma = 1/\tau$ in a system with damping).

Nothing here requires $A$ to be a Hamiltonian — a graph Laplacian, a Markov generator, or a random symmetric matrix all have the same response structure. That universality is why the same machinery describes Casida excitations, Liouville–Lanczos absorption spectra, the kernel polynomial density of states, and nuclear strength functions.

Experiment

Random symmetric $8\times8$ matrix, unit probe $v_0$, $\eta = 0.12$ (the worked example on the continued-fraction page). Computed $A(\omega)$ two ways — directly as $-\operatorname{Im}G/\pi$ and as the explicit Lorentzian sum — they are bit-identical, and $\int A(\omega),d\omega = 1$ (the sum rule). A Lorentzian fit to the isolated top peak returned center $5.015$ (eigenvalue $5.017$), width $0.148$ ($\eta = 0.12$), area $0.014$ (overlap weight $0.011$) — all three parameters recovered. Reproduced by scripts/gen_resolvent_spectral.py.

What would falsify it

Degenerate or near-degenerate eigenvalues merge into a single peak that a single Lorentzian misfits (you need a sum), and a strongly frequency-dependent self-energy (interactions) distorts the lineshape away from Lorentzian — the peak then carries a shifted, asymmetric, finite-lifetime structure. The clean “one Lorentzian per eigenvalue” picture is exact only for a fixed (frequency- independent) operator.