Graduate Can imaginary position operators explain real eigenvalues in quantum mechanics?

[SeM]

Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral :

\begin{equation}
\langle Bx, x\rangle
\end{equation}

when replaced by:\begin{equation}
\langle Bix, ix\rangle
\end{equation}

or\begin{equation}
\langle Bix, x\rangle
\end{equation}

So using (2) or even (3) I get real (but negative) eigenvalues, instead of complex eigenvalues. I have not found any literature on such an absurd thing, as the imaginary position, however, being the only answer to these operators, I am wondering if anyone can point to some further literature on complex operators and complex eigenvalues in QM and whether (2) and (3) make any sense at all

Thanks

 

[vanhees71]

What's $B$? Without clear definitions, I cannot make any sense of the expressions above. Where does this idea of complex/imaginary position eigenvalues come from?

 

[SeM]

It's an idea that comes from a calculation that sums up only if I use an imaginary position operator. Apparently, it is so non-heard of that I will leave it as it is. It has to do with non-hermiticity, where B (not hermitian) only gives a real value on that integral if the position is imaginary.

Thanks!