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Two decades ago, Wang and Ong [Phys. Rev. A 55, 1522 (1997)] hypothesizedthat the local box-counting dimension of a discrete quantum spectrum shoulddepend exclusively on the nearest-neighbor spacing distribution (NNSD) of thespectrum. In this paper, we validate their hypothesis by deriving an explicitformula for the local box-counting dimension of a countably-infinite discretequantum spectrum. This formula expresses the local box-counting dimension of aspectrum in terms of single and double integrals of the NNSD of the spectrum.As applications, we derive an analytical formula for Poisson spectra andclosed-form approximations to the local box-counting dimension for spectrahaving Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), andGaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOEcases, we compare our theoretical formulas with the published numerical data ofWang and Ong and observe excellent agreement between their data and our theory.We also study numerically the local box-counting dimensions of the Riemann zetafunction zeros and the alternate levels of GOE spectra, which are often used asnumerical models of spectra possessing GUE and GSE spacing statistics,respectively. In each case, the corresponding theoretical formula is found toaccurately describe the numerically-computed local box-counting dimension.

Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics