When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)

Question

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!

Answer 1

Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel theorem, has a distribution kernel $\Gamma(x,y)$. It means that your integral formula holds weakly: for $\phi,\psi\in \mathscr D(\mathbb R^n)$ $$ \langle L^{-1}\phi,\psi\rangle_{\mathscr D'(\mathbb R^n),\mathscr D(\mathbb R^n)}= \langle \Gamma(x,y),\psi(x)\otimes\phi(y)\rangle_{\mathscr D'(\mathbb R^{2n}),\mathscr D(\mathbb R^{2n})}. $$ N.B. You wrote $\Gamma(y,x)$ but I prefer the more traditional $\Gamma(x,y)$.

On the other hand every constant coefficient differential operator $P(D)$ (non-zero) has a fundamental solution (Malgrange-Ehrenpreis theorem), and that does not imply invertibility of all these operators: take just $\partial/\partial x_1$ with the huge kernel of distributions of $x_2,\dots, x_n$ and fundamental solution $$ H(x_1)\otimes \delta_0(x_2)\otimes \dots \delta_0(x_n),\quad (H=\mathbf 1_{\mathbb R_+}). $$