User:Eugene Bogomolny/Proposed/Riemann zeros and quantum chaos
Prof. Eugene Bogomolny accepted the invitation on 23 November 2009 (self-imposed deadline: 23 June 2010).
Quantum Potential for Riemann Zeros
For the case of the Riemann zeros Wu and Sprung and others have shown that there exists a potential can be written implicitly in terms of the Gamma function and zeroth-order Bessel function.
\[ f^{-1} (x)=\frac{4}{\sqrt{4x+1} } +\frac{1}{2\pi } \int\nolimits_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) \]
this can be obtained by a direct application of the Riemann-Weil formula
\( \sum\limits_{\gamma }^{}h(\gamma ) =2h\left( \frac{i}{2} \right) -g(0)\ln \pi -2\sum\limits_{n=1}^{\infty }\frac{\Lambda (n)}{\sqrt{n} } g(\ln n)+\frac{1}{2\pi } \int\nolimits_{-\infty }^{\infty }dsh(s)\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{is}{2} \right) \)
to the function \( f^{-1}(x)= 2\sqrt \pi \frac{d^{1/2}}{dx^{1/2}}N(x) \), here N(x) is teh Eigenvalue staircase for the square of the Riemann zeros \( \sum_{n=0}^{\infty} H(x- \gamma^{2} _{n})= \frac{1}{\pi}arg\xi(1/2+i\sqrt{x})=N(x) \)
this potential defined implicitly is just the potential of a certain Hamiltonian in units where \( \hbar = 2m=1 \) so the Energies of the Hamiltonian
\( -\frac{d^{2}}{dx^{2}}y(x)+f(x)y(x)=E_{n}y(x) \) with boundary conditions \( y(0)=0=y(\infty) \) are the square of the Riemann zeros \( E_{n}=\gamma_{n}^{2} \)
the condition \( f^{-1}(x)= 2\sqrt \pi \frac{d^{1/2}}{dx^{1/2}}N(x) \) is obtained for a one dimensional system from the Bohr-Sommerfeld quantization condition
\( \oint p.dx= 2\pi (n+1/2)\hbar \) with the momentum defined by \( p(x)= \sqrt{2m(E_{n}-f(x))} \) , this Bohr-sommerfeld condition may be used to define a Abel type integral equation for the derivative of the inverse of the potential function \( \frac{df^{-1}(x)}{dx} \)
Asymptotics
For big x if we take only the smooth part of the eigenvalue staircase \( N(E)\sim \frac{\sqrt{E} }{2\pi } \log \left( \frac{\sqrt{E} }{2\pi e} \right) \) , then the potential as \( |x| \to \infty \) is positive and it is given by the asymptotic expression \( f(-x)=f(x)\sim 4\pi^2 e^2 \left( \frac{2\epsilon \sqrt{\pi } x+B}{A(\epsilon )} \right) ^{\frac{2}{\epsilon } } \) with \( A(\epsilon )=\frac{\Gamma \left( \frac{3+\epsilon }{2} \right) }{\Gamma \left( 1+\frac{\epsilon }{2} \right) } \) and \( B= A(0) \) in the limit \( \epsilon \to 0 \). This potential is approximately a Morse Potential with \( 16\pi^{2} e^{8|x|} \)
The asymptotic of the energies depend on the quantum number n as \( E_n = \frac{4\pi^2 n^2}{W^2(ne^{-1})} \) here W is the Lambert function
References
- Hua Wu and D. W. L. Sprung, "Riemann zeta and a fractal potential", Physical Review E 48 (1993) 2595.
- G. Sierra, A physics pathway to the Riemann hypothesis, arXiv:math-ph/1012.4264, 2010.
- Fractal fits to Riemann zeros P B Slater, Canadian Journal of Physics, 2007, 85(4): 345–357, 10.1139/p07-050
- Rev. Mod. Phys. 83, 307–330 (2011) Colloquium: Physics of the Riemann hypothesis
- Trace formula in noncommutative geometry and the zeros of the Riemann zeta function Alain Connes http://arxiv.org/pdf/math/9811068v1.pdf
- Phys. Rev. E 51, 6323–6326 (1995) Fractal potentials from energy level
- SIAM J. Math. Anal., 23(2), 482–504. (23 pages) The Recovery of Potentials from Finite Spectral Data http://dx.doi.org/10.1137/0523023
- Some remarks on the Wu-Sprung potential. Preliminary report Diego Dominici
- http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/NTfractality.htm
- http://www.vixra.org/pdf/1301.0078v2.pdf
- http://vixra.org/pdf/1206.0069v5.pdf
