Exercise 39, § 5

Let $n\geqslant 1$ be an integer. Endow $\mathbf{R}^n$ with Lebesgue measure.

Let $v$ be a positive locally square-integrable function on $\mathbf{R}^n$. Denote by Δ the Laplacian on $\mathscr{D}(\mathbf{R}^n)$ or on $\mathscr{D}'(\mathbf{R}^n)$, defined by

Let D be the partial operator on $L^2(\mathbf{R}^n)$ with domain $\mathscr{D}(\mathbf{R}^n)$ such that

a) The partial operator D is symmetric and positive.

b) Let $u\in$ dom($D^*$) be such that $D^*u+u= 0$. Then $\Delta (u) +u= 0$, where the Laplacian of $u\in L^2(\mathbf{R}^n)$ is computed in the sense of distributions.

c) The distributions $\Delta (u)$ and $(v+1)u$ are locally integrable functions, and the distribution $-\Delta (u)$ is positive. (Use exercise 32 of IV, p. 344.)

d) One has $u= 0$. (Approximate $|u|$ by infinitely differentiable functions $w_{\varepsilon}$ by convolution in such a way that $\Delta (w_{\varepsilon})\geqslant 0$ and apply the preceding question.)

e) The partial operator D is essentially self-adjoint.