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my proof is taking normalized Eigenfunctions we would have that:

[tex](<\phi_{n}|H|\phi_{n}>)*=<\phi_{k}|H|\phi_{k}> [/tex]

so the expected value of T is always real,then we would have the identity with the complex part b(x) of the potential:

[tex]\int_{-\infty}^{\infty}dx(|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)=0 [/tex]

for every k,and n so necessarily b=0 so the potential is real and all the eigenfunctions would be real.