Wu–Sprung potential explained

H = p ^{2} + f(x) with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies E _{n} and the potential f(x) .$$\backslash oint\; p\; \backslash ,\; dq\; =2\; \backslash pi\; n(E)=\; 4\backslash int\_0^a\; dx\; \backslash sqrt$$here is a classical turning point so E = f(a) = f( - a) , the quantum energies of the model are the roots of the Riemann Xi function $\backslash xi\; =\; 0$ and f(x) = f( - x) . In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by f ^{-1} (x) = \sqrt \pi (d ^{1/2})/(dx ^{1/2}) N(x) ; with being the eigenvalue staircase $N(x)\; =\; \backslash sum\_^\backslash infty\; H(x\; -\; E\_)$ and is the Heaviside step function.

For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth-order Bessel function.

$$f^\; (x)=\backslash frac\; +\backslash frac\; \backslash int\_^\backslash frac\; \backslash left(\backslash frac\; \backslash left(\backslash frac\; +\backslash frac\; \backslash right)\; -\backslash ln\; \backslash pi\; \backslash right)\; -\backslash sum\backslash limits\_^\backslash infty\; \backslash frac\; J\_0\; \backslash left(\backslash sqrt\; \backslash ln\; n\backslash right)$$

and that the density of states of this Hamiltonian is just the Delsarte's formula for the Riemann zeta function and defined semiclassically as

(1)/(\sqrt \pi) (d ^{1/2})/(dx ^{1/2}) f ^{-1} (x) = \sum ^{inf ty} _{n =0} \delta (x - E _{n} )

here they have taken the derivative of the Euler product on the critical line $\backslash frac+is$; also they use the Dirichlet generating function $\backslash frac=\; -\backslash sum\_^\; \backslash Lambda\; (n)\; e^$.

Λ (n) is the Mangoldt function.

The main idea by Wu and Sprung and others is to interpret the density of states as the distributional Delsarte's formula and then use the WKB method to evaluate the imaginary part of the zeros by using quantum mechanics.

Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function

(\xi(s))/(\xi( 0 )) = (\det(H - s( 1- s) + (1)/() 4))/(\det(H + (1)/() 4))

The main idea inside this problem is to recover the potential from spectral data as in some inverse spectral problems in this case the spectral data is the Eigenvalue staircase, which is a quantum property of the system, the inverse of the potential then, satisfies an Abel integral equation (fractional calculus) which can be immediately solved to obtain the potential.

Asymptotics

For large x if we take only the smooth part of the eigenvalue staircase $N(E)\; \backslash sim\; \backslash frac\; \backslash log\; \backslash left(\backslash frac\; \backslash right)$, then the potential as

|x| \to inf ty is positive and it is given by the asymptotic expression f( - x) = f(x) \sim 4 \pi ^{2} e ^{2} \left( (2 \epsilon \sqrt{\pi)/(x + B}{A(\epsilon) )} \right) ^{2} with A(\epsilon ) = (\Gamma{\left( (3+ \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 as $E\_n\; =\; \backslash frac$, where is the Lambert W function.

References

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• Some remarks on the Wu–Sprung potential. Preliminary report Diego Dominici
• http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/NTfractality.htm