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

zhaiyujia
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[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
 
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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?
 
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
 
zhaiyujia said:
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.
 
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...
 
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