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Wu–Sprung potential

In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies and the potential . here a is a classical turning point so , the quantum energies of the model are the roots of the Riemann Xi function and . In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by ; with N(x) being the eigenvalue staircase and H(x) is the Heaviside step function.

Asymptotics
For large 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) ^{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 as E_n = \frac{4\pi^2 n^2}{W^2(ne^{-1})} , where is the Lambert W function. == References ==
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