Are Finite Well Bound States Enough for a Complete Basis in Quantum Mechanics?

[lttung]

Dear Physics Forum,

I have a question about quantum mechanics.
I know that the solutions of a Hamiltonian will form a complete basis.

However, for a case of a finite well, in the region of bound states (E<0), the number of eigenfunctions is finite, I wonder that they are enough to form a complete basis or not?

For the case of infinite square well or Hydrogen potential, the basis is complete.
But for the case of an attractive Delta well, or a finite well, do the bound states form a complete one, or we need to include the wave function for E>0?

Thank you very much.

Bests,

Tung LE

 

[TMFKAN64]

A finite set of functions cannot be complete. So yes, you need to include E>0 eigenfunctions as well.