When do quadratically integrable functions vanish at infinity?

Mathematics Asked by Dave Lunal on October 25, 2020

In quantum mechanics we use quadratically integrable functions ($$psi in L^2$$).
This means
$$int_{-infty}^infty |psi(x)|^2 mathrm{d}x < infty.$$

I’m interested in the question when those function vanish at infinity, i.e.
$$lim_{x rightarrow pm infty} psi(x) = 0.$$

I know that this is not the case for every function in $$L^2$$, see for example this answer or this answer.

I found in a similar question something interesting:

Suppose $$f : mathbf R to mathbf R$$ is uniformly continuous, and $$fin L^p$$ for some $$pgeq 1$$. Then $$|f(x)|to 0$$ as $$|x| to infty$$.

Another interesting answer is this one.

My questions are:

1. How can one prove the given statement?
2. What are other cases where quadratically integrable functions vanish at infinity?
3. Which cases are relevant in physics (for quantum mechanics)?

Edit:

My remaining question is:

What criteria (beside uniform continuity) do exist, so that quadratically integrable functions vanish (or not) at infinity?

A trivial sufficient condition is that $$f$$ is absolutely continuous with $$f'in L_1(mathbb R)$$. Indeed, the absolute continuity means that $$f(x)=f(0)+int_0^xf'(t),dttext,$$ and $$f'in L_1(mathbb R)$$ implies that $$lim_{xtopminfty},int_0^xf'(t),dt$$ exist which can be seen from a Cauchy-like criterion, observing that $$lim_{Ntoinfty},int_{mathbb Rsetminus[-N,N]}lvert f'(t)rvert,dt=0.$$ (The latter follows for instance from Lebesgue's dominated convergence theorem.)

Note that $$f'in L_1(mathbb R)$$ does even for continuous $$f'$$ not imply the boundedness of $$f'$$ and thus also does not imply that $$f$$ is uniformly continuous.

On the other hand, the assumption $$fin L_p(mathbb R)$$ would be used here only to verify that the limits are not different from zero...

Answered by Martin Väth on October 25, 2020

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