T(f) =  \integal _{-\infty}^x   y f(y)  dy

What are the eigenvalues for T ?

Which eigenvalues lead to square-integrable functions?

I tried differentiating both sides of

T(f) = cf  

and obtained a differential equation

cf'(x)  - x f(x)  = 0,

equivalently, f'/f = x/c

provided c is not 0, in which case f will have the form

 Ke^{x^2/2c}

I'm not terribly sure how to obtain the eigenvalues directly, though.

Thanks in advance for any help!

Cheers,