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For reference, I am reading Lemma 2.2 here.

Suppose that $D\subseteq \mathbb{R}^n$ is non-empty bounded domain with piece-wise smooth boundary. Let $(u_j)$ be a sequence of Dirichlet eigenfunctions on $D$ with corresponding eigenvalues $$ 0 < \lambda_1^2\leq \lambda_2^2 \leq \dots. $$ For each $j\in\mathbb{N}$, define $$ h_j = \frac{1}{\lambda_j}. $$ Fix a function $a \in S(T^\ast D)$ and let $A = Op_{h_j}(a)$ be the quantization of $a$. We denote by $U(t;h_j)$ the Schrödinger propagator. That is, $$ U(t;h_j) = \exp\left({-\frac{itP_j}{h_j}}\right) $$ where $P_j = \frac{1}{2}\left(h_j^2\Delta -1\right)$.

As in the document linked above, we are able to make the following observation $$ \langle Au_j, u_j \rangle_{L^2} =\langle U(-t;h_j) A U(t;h_j) u_j, u_j \rangle_{L^2} =\langle Op_{h_j}(a\circ \varphi_t)u_j, u_j \rangle_{L^2} + \color{red}{O_t(h_j)} $$ where the last step follows from Egorov's theorem and $\varphi_t$ denotes the geodesic billiard flow.

The big-O notation means that our hidden term is bounded above by $C_t h_j$ for all $h_j$ sufficiently small. Can the constants $C_t$ be bounded uniformly in $t$? Also, since the bound only holds for $h_j$ small, does the 'how small' depend on $t$?

We then "average both sides over $t\in [0,T]$". I'm assuming this means integrating over $t\in [0,T]$ and dividing by $T$. This should yield $$ \langle Au_j, u_j \rangle_{L^2} = \color{red}{\langle Op_{h_j}\left(a_T\right) u_j, u_j \rangle_{L^2}} + {O_T(h_j)} $$ where $$ a_T(x, \omega) = \frac{1}{T}\int_0^T a\circ\varphi_t(x, \omega) \,\mathrm{d}{t}. $$

Is the term $\langle (a_T u_j, u_j \rangle_{L^2}$ obtained after changing the order of integration? If so, what integral representation are we using and how does one justify this application of the Fubini-Tonelli theorem?

Am I possibly misunderstanding anything? Any help is appreciated!

Quoka
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  • Good question. It seems these are only sketch of proofs though. Your $a$ here are smooth and compactly supported, so I suppose at some point you have to prove that some smoothness and decay property are transported by your flow (and the same to get the $O_t(h)$. – LL 3.14 Apr 23 '20 at 18:38
  • @LL3.14 Thank you! I also feel as though these are more sketches than proofs. I was unfortunately unable to find anything significantly more complete. If you are aware of a good reference please let me know! (I am aware of Zworski's book, where he does something similar in chapter 15) – Quoka Apr 24 '20 at 00:09
  • As a general rule of thumb, in the case of semiclassical analysis, all the important estimates are completely explicit, with constants depending on suitable C^p/S^p norms of your symbols (and it's probably even easier to get estimates when your symbol is compactly supported). So for your first part, for each $t$ in a compact subset, you apply the Egorov theorem to $a \circ \varphi_t$ -- with thus an explicitly controlled error term in terms of norms of $a \circ \varphi_t$ (continued below) [Note: Zworski's book has the estimates and proofs in Chap 4 and sketches in Chap 14]. – Aphelli Apr 25 '20 at 17:11
  • Then the idea is that for $t$ in a compact subset, the sup of these norms is bounded because $(x,\xi,t) \longmapsto a \circ \varphi_t(x,\xi)$ is smooth compactly supported. So if $|t| \leq T$, yes, you have a uniform bound on the $C_t$. I think all the bounds hold as soon as $h \leq 1$ (because of explicitness). For the second part, yes, you simply change the order of integration. Because $a$ has compact support, once you break down your quantization into honest integrals, it becomes an interversion of integrals where the integrand is continuous compactly supported so it always works. – Aphelli Apr 25 '20 at 17:14
  • @Mindlack How do I express quantization with integration? Wha't formula can be used when the functions isn't defined on all of $\mathbb{R}^n$? Simply extending the function to be $0$ outside of ourt domain? – Quoka Apr 26 '20 at 16:08

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