How to find eigenvalues and eigenfunction

[Another]

defind $\hat{A}f(x)=f(-x)$ find eigenfunction and eigenvalue

I think

$\frac{d}{dx} ( \hat{A}f(x) ) = \frac{d}{dx} f(-x)$
$\hat{A} \frac{d}{dx}f(x) + f(x) \frac{d}{dx} \hat{A} = -\frac{d}{dx} f(x)$
$\hat{A} \frac{d}{dx}f(x) + \frac{d}{dx} f(x) = -f(x) \frac{d}{dx} \hat{A}$
$(\hat{A} + 1)\frac{d}{dx} f(x) = -f(x) \frac{d}{dx} \hat{A}$

multiply by $dx$

$(\hat{A} + 1)d{f(x)} = -f(x) d{ \hat{A} }$
$∫ \frac{1}{f(x)}d f(x) = - ∫ \frac{1}{\hat{A} + 1}d \hat{A}$
$\ln{f(x)} = -\ln{(\hat{A}+1)}+\ln{c}$
$\ln{f(x)} = \ln(\frac{c}{\hat{A}+1})$

so...
$f(x) = \frac{c}{\hat{A}+1}$ i think it's wrong

 

[stevendaryl]

Why are you taking derivatives?

The definition of an eigenfunction of an operator is:

$\hat{A} f(x) = \lambda f(x)$

where $\lambda$ is a number.

 

[bhobba]

The answer seems to have no relation to the question ie where does the derivative come from. As stated its an interesting question with an interesting answer. Expand f(x) in a power series and write the eigenvalue equation (f(-x) = a*f(x) - a eigenvalue - f(x) eigenfuntion) - then equate terms of the same power.

 

[ehild]

defind $\hat{A}f(x)=f(-x)$ find eigenfunction and eigenvalue

Apply the operator twice - what do you get? What is $\hat{A}(\hat{A}f(x))$? So what are the eigenvalues of $\hat{A^2}$? What can be the eigenvalues of $\hat{A}$?