How to find eigenvalues and eigenfunction

OP warned about not using the homework template
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
Staff Emeritus
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 and Another
bhobba
Mentor
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.

Another
ehild
Homework Helper
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}##?

bhobba, Another, vela and 1 other person