-- Eigenvalue problem for sufficiently behaved functions

[k.hofmann]

Suppose f is a sufficiently well-behaved function (say, smooth function with compact support) defined on the set of real numbers, and define

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,

K. H.