Exercise 14, § 5

Let $n\in \{1,2,3\}$. Denote by Δ the Laplace operator on $L^2(\mathbf{R}^n)$, and by D its domain. It is a positive partial operator.

a) There exists a real number $c\geqslant 0$ such that for every $f\in D$, one has

b) For every real number $a >0$, there exists a real number $b\geqslant 0$ such that for every $f\in D$, one has

(Apply the preceding question to $x\mapsto f(\alpha x)$, where $\alpha \in \mathbf{R}^*_+.$)

c) Let $g\in \mathscr{L}^{\infty}(\mathbf{R}^n) +\mathscr{L}^2(\mathbf{R}^n)$. If $g$ is real-valued, then the partial operator $\Delta +m_g$ is self-adjoint.

d) Let $\alpha$ be a real number such that $0\leqslant \alpha < n/2$. Let N be the euclidean norm on $\mathbf{R}^n$ and $g$ the function on $\mathbf{R}^n$ defined by $g(x) = 1/N(x)^{\alpha}$ if $x\not = 0$, and $g(0) = 0$. The partial operator $\Delta +m_g$ is self-adjoint.