# Why should fields in AdS spacetime vanish at infinity, but not in Minkowski spacetime?

I was watching the following lectures by Prof. Ashoke Sen. Between 39:00 and 56:00, he was solving the equation of classical field in the AdS global coordinates, and says that the values of $$\omega$$ are discrete to have regular solutions at $$r=0$$ and at $$r=\infty$$. One could write the scalar field equation in Minkowski space-time using polar coordinates, and a similar analysis of the solution may yield the result that the energy of the field comes in discrete values like AdS. But solving the equation in the Euclidean coordinates, after decomposing the field into Fourier components, shows that possible values of $$\omega$$ vary continuously. Where is the discrepancy? According to me, it is related to the fact that the Fourier components diverge at infinity, unlike the AdS scalar field which isn't allowed to (diverge). I remember reading that the AdS spacetime behaves like a box with the discreteness of the energy values being one of the reasons. So, why is it natural to have a non-diverging solution at spatial infinity in AdS spacetime but not in Minkowski?