Rigged Hilbert Space Φ ⊂ H ⊂ Φ'

[strangerep]

Another idea to remove for the difficulty is modification of energy eigenfunction; [...]

It is usually more profitable to study thoroughly what other people have already achieved before constructing one's own theory.

In this vein, it may be helpful to study distribution theory more deeply. I found Appendix A of Nussenzveig's book on "Causality and Dispersion Relations" quite helpful.

 

[sweet springs]

Thanks for your comment.
I am excited to know that the square well potential that appears n many QM textbooks seems to have something more to be explored.
I agree with you that I should learn more.