Relation between potential and bound states in Schrodinger equation.

[arroy_0205]

Suppose I have Schroedinger equation in the form:
$$-u''(x)+V(x)u(x)=Eu(x)$$
The potential is such that as |x| -> Infinity, V(x) reaches a constant positive value. In this case can we have bound state/plane wave solutions for u(x) with E > 0 ?

 

[Meir Achuz]

You can have bound state solutions with E>0.
Just take any V that-->0 and add any positive constant to it.
You can't have plane wave solutions if V depends on x.