Find eigenvalues for $T(f) = \int_{-\infty}^x tf(t)dt$

Question

Let $V$ be the linear space of all functions continuous on $(-\infty, \infty)$ and such that that the integral $\int_{-\infty}^x tf(t)\,dt$ exists. If $f \in V$, let $g=T(f)$ be defined as $g(x) = \int_{-\infty}^x tf(t)\,dt$. Prove that every $\lambda < 0$ is an eigenvector and determine the eigenfunctions corresponding to $\lambda$.

We know that $T(f) = \int_{-\infty}^x tf(t)\,dt = \lambda f(x)$. (Not sure if this is right so far.) So know what do I do?

Answer 1

Integral equations are pretty difficult to solve directly. On the other hand, we can easily solve differential equations. Try taking a derivative and use the fundamental theorem of calculus to get a differential equation which you can easily solve. (You should have $\lambda f(x)$, not $\lambda f(t)$ on the right hand side, by the way.)

Answer 2

as cameron suggests, take the equation $$ \int_{-\infty}^x tf(t)\, dt = \lambda f(x) \tag 1$$

first, we will deal with the case $\lambda = 0.$ diffrentiang $(1)$ gives $xf(x) = 0$ implying $f \equiv 0$ contradicting that $f$ is an eigenfunction.

now we are assuming $\lambda \neq 0.$ differentiating $(1),$ we get $$ xf(x) = \lambda f'(x).$$ separate the variables to get $$\frac{df}{f} = \frac 1\lambda x\, dx $$ gives $$f = Ce^{x^2/\lambda} $$ now the boundary condition $\lim_{x \to -\infty}f(x) = 0$ implies $\lambda < 0.$

Answer 3

You wrote "We know that $T(f) = \int_{-\infty}^x tf(t)\,dt = \lambda f(x)$."

You could say we know that $T(f) = \int_{-\infty}^x tf(t)\,dt$ for every $f$ in the domain, and we know that if $f$ is an eigenfunction with eigenvalue $\lambda$ then the second equality holds. That doesn't indicate that any such function exists.

Differentiating both sides of the equation $$ \int_{-\infty}^x tf(t)\,dt=\lambda f(x) $$ we get $$ xf(x) = \lambda f'(x) $$ so that $$ \frac x \lambda\,dx = \frac{df} f. $$ Hence $$ \frac{x^2}{2\lambda} + C = \log |f(x)|. $$ $$ Ae^{x^2/(2\lambda)} = f(x). \tag 1 $$ This holds if $f$ is an eigenfunction with eigenvalue $\lambda$, so we've shown that if any such eigenfunction exists, then it looks like $(1)$.

Now ask yourself: For which values of $\lambda$ is it the case that $\displaystyle\int_{-\infty}^x t e^{t^2/(2\lambda)} \, dt$ exists for every $x$?