I'm solving a quantum mechanics problem and I've got the following integral as a result
$$ \int^{\infty}_{-\infty} \, \delta \left(x-\frac{L}{2}\right) \, \left(x-x_0 \, \frac{\partial}{\partial x} \right)^n \, e^{-(x/x_0)^2/2} \, dx. $$
It should be pretty straightforward to evaluate, thanks to the delta function, however I can't due to the differential operator. My professor said there was a way to get rid of the operator, but couldn't remember it.
The essence of the problem is:
Using an operator form of the Hermite polynomials, $$ H_{n}(y) = e^{y^2/2} \, \left(y - \frac{d}{dy}\right)^n \, e^{-y^2/2}. $$ Letting $y = x/a$ leads to $$ a^n \, e^{-x^2/(2 a^2)} \, H_{n}\left(\frac{x}{a}\right) = \left(x - a^2 \, \frac{d}{dx}\right)^n \, e^{-x^2/(2 a^2)} $$ and $$ \int_{-\infty}^{\infty} \, \delta \left(x-\frac{L}{2}\right) \, \left(x - a^{2} \ \frac{\partial}{\partial x} \right)^n \, e^{-(x/a)^2/2} \, dx = a^{n} \, e^{-L^2/(8 a^{2})} \, H_{n}\left(\frac{L}{2 a}\right). $$