Is \(\psi(x) = \frac{1}{x^{\alpha}}\) in Hilbert Space?

[zhaiyujia]

[SOLVED] Hilbert Space

Homework Statement

For What Values of $$\psi(x)=\frac{1}{x^{\alpha}}$$ belong in a Hilbert Sapce?

Homework Equations

$$\int x^{a}=\frac{1}{a+1} x^{a+1}$$

The Attempt at a Solution

I tried to use the condition that function in Hilbert space should satisfy:
$$\int\psi^{2}=A$$ but it seems always infinite exist in x=0 or x=infinite

 

[Gokul43201]

Why are you writing the integral of x^a, when you want to examine the integral of 1/x^{2a}?

Also, have you written down the question completely? What is the domain on which $\psi(x)$ is defined?

 

[zhaiyujia]

one is alpha and another is a. I just write a integral equation, a = 2*alpha. I think it is not the key point. The question is complete. I asked my professor if there is some constrain of x, she said x can be any value. that is to say the integral will from minus infinite to infinite

 

[Gokul43201]

one is alpha and another is a.

Okay, they both looked the same to me.

The question is complete.

It can not be. There's at least a couple of missing words. Here's one way to write a somewhat more complete question:

For what values of $\alpha$ does $\psi(x)=1/{x^{\alpha}}$ belong in a Hilbert space?

I asked my professor if there is some constrain of x, she said x can be any value. that is to say the integral will from minus infinite to infinite

This needs to be specified in the question. You have not completely specified a function unless you describe its domain.

 

[zhaiyujia]

Thanks, I explained it in the interval of minus infinite to minus zero and zero to infinite. I guess a wave function with singularity is not a good one in physic...