How to invert the following operators with frequency and position based multipliers?

Question

I am working on a problem in which I have operators of the following form.

$$\hat{H} = \frac{d}{dx} + x$$

One thing that would really help my analysis of this particular problem was if I was able to construct an analytical representation of the inverse of this operator. As an additional question. Lets say that I have an operator that consists of some two operators that act as a Fourier multiplier and a position based multiplier.

$$\hat{A} = g(k) + f(x)$$

Where g(k) represents the Fourier multipler component of the original operator. For example in my prior example this would become the following.

$$\hat{H} = ik + x$$

Is it possible to construct an inverse for the operator $\hat{A}$ given that I know the function inverses of g(k) and f(x).

Thank you for the help and I hope that my lack of rigor is fine in defining the problem.

Answer 1

Your operator is not one-to-one, it maps the (rescalled) Gaussian $f(x) = e^{-\frac{x^2}2}$ to the identically zero function.

Answer 2

In some sense you want $K$ such that

$$HK = \delta_0$$

The $K$ that satisfies this is (convolution by) $K = e^{-\frac{x^2}{2}}\theta(x)$ where $\theta$ is the Heaviside step function. The inverse, however, is only applicable on the subspace that excludes the nullspace of $H$. This gives your particular solution, and adding the nullspace component gives the homogeneous solution.

If you are curious to learn more, $K$ is known as the fundamental solution of $H$.