# is it possible to have different states other than just one in Dirac Delta Potential?

Let's say, initially the state is in first excited state of finite well potential and then I change the width & depth of the well, eventually to Dirac delta potential, then what happens to the state of the wave function?

(You could try to make the well deeper as you make it narrower in such a way that the first excited state will stay at a constant energy depth below the continuum. That won't work, though, because the ground-state energy will sink to $-\infty$, and it won't give you a well-defined operator as a limit.)