If I have a Schrodinger Equation, in 4d space, of the form
$$0=\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\partial{x^2}}+\frac{\partial^2\Psi}{\partial{y^2}}+\frac{\partial^2\Psi}{\partial{z^2}}+\frac{\partial^2\Psi}{\partial{w^2}}\right)+A\Psi$$
and set $w=ict$ and $A=-mc^2/2$
then I get
$$0=\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\partial{x^2}}+\frac{\partial^2\Psi}{\partial{y^2}}+\frac{\partial^2\Psi}{\partial{z^2}}+\frac{\partial^2\Psi}{(ic)^2\partial{t^2}}\right)-\frac{mc^2}{2}\Psi$$
which is equivalent to the Klein Gordon Equation. This means that the Klein Gordon Equation is mathematically the same as a variant of The time-independent Schrodinger Equation in 4 dimensional spatial position space with one of the coordinates being imaginary.
In the time-independent Schrodinger Equation there is a potential operator, which is a function of position space, not a wavefunction, or derivative of a wavefunction, and which describes the interaction between particles. I was wondering if there is something analogous to the potential operator in relativistic quantum mechanics, in the sense of being a function of spacetime, and being neither a wavefunction nor a derivative of a wavefunction. For instance in relativistic quantum mechanics is there an equation of the form
$$B\Psi=\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\partial{x^2}}+\frac{\partial^2\Psi}{\partial{y^2}}+\frac{\partial^2\Psi}{\partial{z^2}}+\frac{\partial^2\Psi}{(ic)^2\partial{t^2}}\right)+L\Psi$$ with $L$ being a function of spacetime and describing an interaction?
