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Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral
Jamal Sakhr1, John M Nieminen2
1Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7.
Physical Review. E
|May 20, 2018
Summary
We derived a formula showing that the local box-counting dimension of quantum spectra depends on nearest-neighbor spacing distributions (NNSD). This formula accurately predicts dimensions for various spectra, including the Riemann zeta function zeros.
Area of Science:
- Quantum mechanics
- Mathematical physics
- Spectral theory
Background:
- The local box-counting dimension of quantum spectra is a measure of spectral complexity.
- A previous hypothesis suggested this dimension depends solely on the nearest-neighbor spacing distribution (NNSD).
Purpose of the Study:
- To validate the hypothesis that the local box-counting dimension depends exclusively on the NNSD.
- To derive an explicit formula for this dimension.
Main Methods:
- Derivation of an explicit formula for the local box-counting dimension using integrals of the NNSD.
- Analytical derivation for Poisson spectra.
- Approximation formulas for Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spectra.
- Numerical analysis of Riemann zeta function zeros and GOE spectra.
Main Results:
- An explicit formula for the local box-counting dimension was derived, dependent on the NNSD.
- Analytical formulas for Poisson and approximations for GOE, GUE, and GSE spectra were obtained.
- Excellent agreement was found between theoretical predictions and published numerical data for Poisson and GOE spectra.
- Numerical studies confirmed the accuracy of the derived formulas for Riemann zeta function zeros and GOE spectra.
Conclusions:
- The hypothesis regarding the dependence of the local box-counting dimension on NNSD is validated.
- The derived formulas provide accurate predictions for various quantum spectra.
- This work offers a new theoretical tool for analyzing spectral properties.
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