Exercise 18, § 5

Let $n$ be an integer $\geqslant 1$. Let D denote the partial operator on $L^2(\mathbf{T}^n)$ with domain $\mathscr{C}^{\infty}(\mathbf{T}^n)$ such that

for every $\varphi \in \mathscr{C}^{\infty}(\mathbf{T}^n)$,

a) The partial operator D is essentially self-adjoint. Let Δ denote its closure.

b) The partial operator Δ is self-adjoint and positive.

c) The partial operator Δ has compact resolvent.

d) Let $\lambda \in \mathbf{R}$. One has $\lambda \in$ Sp(Δ) if and only if there exist natural integers $k_1, . . .,k_n$ such that $\lambda = 4\pi^2(k^2_1+\cdots +k^2_n)$.

e) For every eigenvalue $\lambda$ of Δ, let $N(\lambda )$ be the dimension of the corresponding eigenspace. One has

when $T\rightarrow +\infty$.

f) If $n\geqslant 2$, then lim sup$_{\lambda}N(\lambda ) = +\infty$. (In the case $n= 2$, consider $\lambda = 4\pi^2m$, where $m$ is a product of prime numbers of the form $4k+ 1.$) g) If $n= 1$ or $n= 2$, one has

Card(Sp($u$)$\cap [0,T]$) $=o(T)$,

and if $n\geqslant 3$, one has

lim inf$_{T\rightarrow+\infty}\frac{1}{T}$ Card(Sp($u$)$\cap [0,T]$)$>0$.