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)?


My first question was answered in the comments by @reuns.

My remaining question is:

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

One Answer

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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