Concept in mathematical physics
In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian formula_1 with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies formula_2 and the potential formula_3.
formula_4
here a is a classical turning point so formula_5, the quantum energies of the model are the roots of the Riemann Xi function formula_6 and formula_7. In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by formula_8; with "N"("x") being the eigenvalue staircase formula_9 and "H"("x") 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.
formula_10
and that the density of states of this Hamiltonian is just the Delsarte's formula for the Riemann zeta function and defined semiclassically as formula_11
formula_12
here they have taken the derivative of the Euler product on the critical line formula_13; also they use the Dirichlet generating function formula_14. formula_15 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 formula_16
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 formula_17, then the potential as formula_18 is positive and it is given by the asymptotic expression formula_19 with formula_20 and formula_21 in the limit formula_22. This potential is approximately a Morse potential with formula_23
The asymptotic of the energies depend on the quantum number n as formula_24, where "W" is the Lambert W function.