Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator $$ Lu:=-\Delta u+V(x)u $$ where $V\geq0$ is the electric potential, and $V\in\mathscr{B}$ where $\mathscr{B}$ is some function space (say, for example, $L^{\infty}_{loc}(\mathbb R^n)$). Now let $f\in C_{c}^{\infty}(\mathbb R^n)$ be arbitrary. Suppose the space $\mathscr{B}$ is such that the equation $$ Lu=f $$ has a unique solution in a weak sense. In this case we can invert the operator $L$ and write $$ u=L^{-1}f $$ for $f\in C_c^{\infty}(\mathbb R^n)$. A locally integrable function $\Gamma(x,y)$ is called the fundamental solution for operator $L$ if it satisfies $$ L\Gamma(\cdot,y)=\delta_y $$ in the sense of distributions. If the operator $L$ is invertible and the fundamental solution exists, then we can write \begin{equation} (L^{-1}f)(x)=\int\limits_{\mathbb R^n}\Gamma(y,x)f(y)\,dy. \end{equation}

I am interested in the following specific questions:

**a)** Does every invertible operator given by a PDE as in this case necessarily have a kernel? If so, is this kernel necessarily the fundamental solution?

**b)** Does existence of the fundamental solution imply invertibility of $L$?

These questions are motivated by an observation that in some research papers, the fundamental solution is simply said to be the "kernel of the operator $L^{-1}$", without any justification that such a kernel exists, or that such a kernel is necessarily the fundamental solution.

Thanks for your attention!

Riesz representation theorem. $\endgroup$ – Igor Khavkine Oct 15 '16 at 20:05