Let define $H_c: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ as a densly defined linear operator as follow: $$ H_c u= \Delta u - \frac{c}{|x|^2} u$$
In reading a paper I encountered to the following theorem
$H_c$ is essentially self adjoint if and only if $c\leq \frac{n(n-4)}{4}$.
My try:
From a well-known theorem it is enough to show that $$ ker((H_c)^* \pm i I) ={0} $$ It means that $$ -\Delta u+ \frac{c}{|x|^2}u \pm i u=0 $$ has the trivial solution iff $c \leq \frac{n(n-4)}{4}$.