Spectral Density Function

The Spectral Density Function is the discrete spectrum of a BPT harmonic system: a line at each harmonic frequency, weighted by that harmonic's energy. Because BPT harmonics sit at quadratic positions and the spacing between them grows linearly with depth, the spectral lines are unevenly placed, and weighting each line by its energy produces a characteristic signature that the book uses to tell BPT harmonic structures from classical ones. The energy weights must fall off fast enough for the total to stay finite, which the Harmonic Energy Distribution expresses as the requirement γ > 3. The formal statement is σ(f) = sum_n δ(f − H_n) E_n, where σ(f) is the Spectral Density Function, δ the Dirac delta function, f the frequency variable, H_n the nth harmonic frequency, E_n the energy of the nth harmonic and n the harmonic index. Dimensions from the book's Where-list: σ(f) [J·s], δ [𝕋], f and H_n [𝕋⁻¹], E_n [J], n [∅]; the book's check is consistent. It connects to Recursive Frequency Spacing, the Convergence Parameter, Amplitude Scaling, Mode Density Function and Harmonic Complexity Scaling.