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:*

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

**Edit:**

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?

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

Get help from others!

Recent Answers

- Peter Machado on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP