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More on volume dependence of spectral weight function

Spectral weight functions are easily obtained from two-point correlation functions and they might be used to distinguish single-particle from multi-particle states in a finite-volume lattice calculation, a problem crucial for many lattice QCD simulations. In previous studies, it is shown that the spectral weight function for a broad resonance shares the typical volume dependence of a two-particle scattering state i.e. proportional to $1/L^3$ in a large cubic box of size $L$ while the narrow resonance case requires further investigation. In this paper, a generalized formula is found for the spectral weight function which incorporates both narrow and broad resonance cases. Within Lüscher's formalism, it is shown that the volume dependence of the spectral weight function exhibits a single-particle behavior for a extremely narrow resonance and a two-particle behavior for a broad resonance. The corresponding formulas for both $A^+_1$ and $T^-_1$ channels are derived. The potential application of these formulas in the extraction of resonance parameters are also discussed.