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[tex] H=\frac{-\hbar^{2}}{2m}D^{2}\phi(x)+V(x)\phi(x) [/tex]

then the RH is equivalent to prove that exist a real potential V(x) of the Hamiltonian so that the values E_n [tex]H\phi=E_{n}\phi [/tex] satisfy the equation [tex]\zeta(1/2+iE_{n})=0 [/tex] that is the roots of the Riemann zeta function are the Energies of the system....

you only need to prove that V(x) is real,so the H will be self-adjoint,we can write the solution of the problem (approximately) by the wave function:

[tex]\phi=e^{iS/\hbar} [/tex] with [tex]S=\int(2mE_{n}-2mV(x))^{0.5}dx [/tex]

with that you can substituting into Schroedinguer equation get a differential equation of second order for V(x) [tex]F(x,V(x),DV(x),D^{2}V(x),E_{n})=0 [/tex]

teh key is that you needn,t solve the equation you only have to prove that the potential will be real by knowing that energies satisfy [tex]\zeta(1/2+iE_{n})=0 [/tex] i will try to submit to some math teacher to see if i can do my PhD in math-physics in this matter....