Square-integrable functions question

[YoungEverest]

Note: This is not a homework problem but merely independent study, if it's in the wrong place, please move!

Homework Statement

I want to know how changing a square integrable function changes the result of an integral. So that if a function is square integrable and you multiply it by x, is it still square integrable?

Homework Equations

If $$\int$$ |f(x)|^2 dx < $$\infty$$

then is $$\int$$ x^2 |f(x)|^2 dx < $$\infty$$ ?

The Attempt at a Solution

I have no idea how to attempt this, I just have a feeling that it is true as multiplying by x should not change the fact that the area under the curve is contained, and not open at +/- infinity.

 

[vela]

What about f(x)=1/√(1+x²)?

 

[YoungEverest]

Ah, I see. Now my task becomes a little trickier. Thanks for the help!

Are there any general rules which apply when modifying square integrable functions? For that would make my task easier.

 

[hunt_mat]

What about f(x)=1/√(1+x²)?

Nice counter example...

 

[hunt_mat]

Ah, I see. Now my task becomes a little trickier. Thanks for the help!

Are there any general rules which apply when modifying square integrable functions? For that would make my task easier.

You could try and see about an algebra of square integrable functions

 

[eczeno]

square-integrable functions form a hilbert space, so any linear combination of s-i functions will also be s-i. however, when it comes to multiplying these functions, i don't know of any way of determining which products of s-i functions will also be s-i (other than just trying the integral). and since, f(x)=x is not s-i, well, i can say even less.

 

[hunt_mat]

What about showing the product of square integrable functions is square integrable or looking at function which decay fast enough.

 

[YoungEverest]

Ah yes Eczeno, you make a good point. I have done a couple of courses on linear algebra so I may go back and look at the conditions for a space and do some more tests. But seeing as a counterexample has been presented I do not see much hope.

Essentially I was trying to apply this to quantum mechanics, and what this shows is that even if the wave function can be normalised, it can still have an undefined standard deviation. What I need to find out is if these functions are actually wave functions, or solutions to Schrödinger's equation. It might just be a pathological case...

Hunt mat, essentially this is what I want to try and prove. But I do not know how!