# Schrödinger equation on a torus

Morrey's Identity:

Define the Torus: $$\pi^{d}=\frac{\mathbb{R}^{d}}{2 \pi \mathbb{Z}^{d}}$$

Let $$m \in \mathbb{N}$$ and $$s\in \mathbb{R}$$ such that $$s > m + \frac{d}{2}$$ . Prove that there exists $$c > 0$$ such that for all $$u \in H^{s}(\pi ^{d})$$ (Sobolev space) we have: $$\|u\|_{C^{m}(\pi ^{d})} \leq c\|u\|_{H^{s}(\pi ^{d})}$$ , for some constant $$c$$.

Remark that : $$H^{s}(\pi ^{d}) \subset C^{m}(\pi^{d})$$ for $$s >m +\frac{d}{2}$$.

Solution :

For $$m=0$$, $$s>\frac{d}{2}$$ we have to prove that there exist for all $$u \in$$ $$H^{s}(\pi ^{d}$$) we have : $$\|u\|_{C^{0}}$$ $$\leqslant$$ c$$\|u\|_{H^{s}(\pi^{d})}$$

Let $$u \in H^{s}(\pi^{d})$$

\begin{aligned} | u ( x ) | & = \left| \sum _ { k \in \mathbb{Z} ^ { d } } \hat { u } ( k ) e_{ k} \right| \\ & \leqslant \sum_ { k \in \mathbb { Z } ^ { d } } | \hat { u } ( k ) | \langle k \rangle ^ { s } \langle k \rangle ^ { - s } \\ & \leqslant \left( \sum_ { k \in \mathbb{Z} ^ { d } } | \hat { u } ( k ) | ^ { 2 } < k > ^ { 2 s } \right) ^ { \frac { 1 } { 2 } } \left( \sum _ { \mu \in x ^ { d } } \langle k \rangle ^ { - 2 s } \right) ^ { \frac{1}{2} } \\ & \leqslant \| u \| _ { H ^ { s }(\pi ^ {d})} \left( \sum _ { k \in \mathbb{Z} ^ { d } } \frac { 1 } { \|k\| ^ { 2s } } \right) ^ { \frac { 1 } { 2 } } \\ & \leqslant c \|u\| _{H^{s}(\pi^{d})} \end{aligned}

where $$^{s}$$ = $$(1+\|k\|^{2})^{\frac{s}{2}}$$ and k$$\in$$ $$\mathbb{Z}^{d}$$

• You have to explain what is $k^{2s}$, and for which $s$ the series $\Sigma _{k\in Z^d}{1\over {\vert\vert k \vert\vert^{2s}}}$ converges. Be carefull that $k$ is a vector, not an number. Jan 31, 2020 at 6:06