Suppose I have a random variable $X_0$ with a p.d.f $f_0$ supported on the real interval $[a_0, b_0]$. $X_1$ is the restriction to $[a_1, b_1]$ of the sum $X_0 + g$, where $g$ is normally distributed $g \sim \mathcal{N}(0,1)$

$$f_1(y) = \frac{\int_{x=a_0}^{x=b_0} f_0(x) e^{-(y-x)^2/2}~dx }{\int_{x'=a_1}^{x'=b_1}\int_{x=a_0}^{x=b_0} f_0(x) e^{-(x'-x)^2/2}~dx~dx'}$$

or $$f_1 = \frac{1}{Z} L( f_0, (a_0,b_0,a_1,b_1))$$ Where $Z$ is a normalizing factor. $$Z = \int_{a_1}^{b_1}L( f_0,(a_0,b_0,a_1,b_1))(x')~dx'$$

$L$ is a linear operator over functions in $\mathcal{L}^2$

What are its eigenvectors? What happens when I replace the interval with a $n$ dimensional box and the normal distribution with a multivariate normal?

Thanks!

Clarification: I'm looking at f in $\mathcal{L}^2(\mathbf{R})$, not $\mathcal{L}^2([a_0,b_0])$ otherwise $L$ is obviously not an endomorphism.