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I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.

I am studying eigenfunctions of the Laplacian (both Dirichlet and Neumann cases) on a specified bounded domain $D\subseteq \mathbb{R}^2$ which has piece-wise smooth boundary. Next, consider the billiard map $$\phi : M \to M,$$ where $M$ is the subset of the unit cotangent bundle $S^\ast(\partial D)$ containing only the inward pointing directions. The billiard map models the behaviour of a free particle in the space $D$. Suppose we are base point $x\in \partial D$ and a unit direction vector $w$ pointing inwards. Then a free particle starting at x and travelling in direction $w$ will eventually hit the boundary $\partial D$. Let $x^\prime$ be the point of incidence and suppose that $w^\prime$ is the new direction of the particle upon hitting the boundary. Then $\phi(x, w) = (x^\prime, w^\prime)$. The Billiard map can also be described in terms of the billiard flow. That is, suppose that $\varphi_t : S^\ast D \to S^\ast D$ is the billiard flow. The billiard flow $\varphi_t$ solves the equation $$ \partial_t\varphi_t(x, \omega) = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\varphi_t(x, w). $$ Then $$ \phi(x, w) = \varphi_{\tau(x,w)}(x, w) $$ where $$ \tau(x, w) = \inf\{t > 0: \varphi_t(x, w) \in M\}. $$

Let $p$ be a given function that is invariant with respect to the billiard map. i.e, $$p:M \to \mathbb{C}$$ is such that $$ p\circ \phi = p \quad \text{on } M. $$ Supposed that the function $p$ is defined on all of $\mathbb{R}^2\times \mathbb{R}^2$. By quantizing $p$, we obtain a pseudo-differential operator $P_h = p(x, hD)$. That is, $$ P_hu(x) = \frac{1}{\left(2\pi h\right)^2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} e^{\frac{i}{h}\langle x-y, \xi\rangle} p(x, \xi) u(y)\,\mathrm{d}{y}\mathrm{d}{\xi} $$ for every sufficiently nice function $u$.

I have shown that $P_h$ shares a complete (in $L^2(D)$) collection of eigenfuntions with the Laplacian $\Delta$. In particular, $P_h$ commutes with the Laplacian $\Delta$.

Is there an intuitive reason why the operator I obtained commutes with the Laplacian? Can anyone provide information regarding the significance (or applications) of such an operator?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.

I am studying eigenfunctions of the Laplacian (both Dirichlet and Neumann cases) on a specified bounded domain $D\subseteq \mathbb{R}^2$ which has piece-wise smooth boundary. Next, consider the billiard map $$\phi : M \to M,$$ where $M$ is the subset of the unit cotangent bundle $S^\ast(\partial D)$ containing only the inward pointing directions. The billiard map models the behaviour of a free particle in the space $D$. Suppose we are base point $x\in \partial D$ and a unit direction vector $w$ pointing inwards. Then a free particle starting at x and travelling in direction $w$ will eventually hit the boundary $\partial D$. Let $x^\prime$ be the point of incidence and suppose that $w^\prime$ is the new direction of the particle upon hitting the boundary. Then $\phi(x, w) = (x^\prime, w^\prime)$. The Billiard map can also be described in terms of the billiard flow. That is, suppose that $\varphi_t : S^\ast D \to S^\ast D$ is the billiard flow. The billiard flow $\varphi_t$ solves the equation $$ \partial_t\varphi_t(x, \omega) = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\varphi_t(x, w). $$ Then $$ \phi(x, w) = \varphi_{\tau(x,w)}(x, w) $$ where $$ \tau(x, w) = \inf\{t > 0: \varphi_t(x, w) \in M\}. $$

Let $p$ be a given function that is invariant with respect to the billiard map. i.e, $$p:M \to \mathbb{C}$$ is such that $$ p\circ \phi = p \quad \text{on } M. $$ Supposed that the function $p$ is defined on all of $\mathbb{R}^2\times \mathbb{R}^2$. By quantizing $p$, we obtain a pseudo-differential operator $P_h = p(x, hD)$. That is, $$ P_hu(x) = \frac{1}{\left(2\pi h\right)^2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} e^{\frac{i}{h}\langle x-y, \xi\rangle} p(x, \xi) u(y)\,\mathrm{d}{y}\mathrm{d}{\xi} $$ for every sufficiently nice function $u$.

I have shown that $P_h$ shares a complete (in $L^2(D)$) collection of eigenfuntions with the Laplacian $\Delta$. We also note that, $P_h$ commutes with the Laplacian $\Delta$.

Is there an intuitive reason why the operator I obtained shares an orthonormal basis of (Dirichlet/Neumann) eigenfunctions with the Laplacian? Can anyone provide information regarding the significance (or applications) of such an operator?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.

I am studying eigenfunctions of the Laplacian (both Dirichlet and Neumann cases) on a specified bounded domain $D\subseteq \mathbb{R}^2$ which has piece-wise smooth boundary. Next, consider the billiard map $$\phi : M \to M,$$ where $M$ is the subset of the unit cotangent bundle $S^\ast(\partial D)$ containing only the inward pointing directions. The billiard map models the behaviour of a free particle in the space $D$. Suppose we are base point $x\in \partial D$ and a unit direction vector $w$ pointing inwards. Then a free particle starting at x and travelling in direction $w$ will eventually hit the boundary $\partial D$. Let $x^\prime$ be the point of incidence and suppose that $w^\prime$ is the new direction of the particle upon hitting the boundary. Then $\phi(x, w) = (x^\prime, w^\prime)$. The Billiard map can also be described in terms of the billiard flow. That is, suppose that $\varphi_t : S^\ast D \to S^\ast D$ is the billiard flow. The billiard flow $\varphi_t$ solves the equation $$ \partial_t\varphi_t(x, \omega) = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\varphi_t(x, w). $$ Then $$ \phi(x, w) = \varphi_{\tau(x,w)}(x, w) $$ where $$ \tau(x, w) = \inf\{t > 0: \varphi_t(x, w) \in M\}. $$

Let $p$ be a given function that is invariant with respect to the billiard map. i.e, $$p:M \to \mathbb{C}$$ is such that $$ p\circ \phi = p \quad \text{on } M. $$ We note that the function $p$ is defined on all of $\mathbb{R}^2\times \mathbb{R}^2$. By quantizing $p$, we obtain a pseudo-differential operator $P_h = p(x, hD)$. That is, $$ P_hu(x) = \frac{1}{\left(2\pi h\right)^2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} e^{\frac{i}{h}\langle x-y, \xi\rangle} p(x, \xi) u(y)\,\mathrm{d}{y}\mathrm{d}{\xi} $$ for every sufficiently nice function $u$.

I have shown that $P_h$ shares a complete (in $L^2(D)$) collection of eigenfuntions with the Laplacian $\Delta$. In particular, $P_h$ commutes with the Laplacian $\Delta$.

Is there an intuitive reason why the operator I obtained commutes with the Laplacian? Can anyone provide information regarding the significance (or applications) of such an operator?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.

I am studying eigenfunctions of the Laplacian (both Dirichlet and Neumann cases) on a specified bounded domain $D\subseteq \mathbb{R}^2$ which has piece-wise smooth boundary. Next, consider the billiard map $$\phi : M \to M,$$ where $M$ is the subset of the unit cotangent bundle $S^\ast(\partial D)$ containing only the inward pointing directions. The billiard map models the behaviour of a free particle in the space $D$. Suppose we are base point $x\in \partial D$ and a unit direction vector $w$ pointing inwards. Then a free particle starting at x and travelling in direction $w$ will eventually hit the boundary $\partial D$. Let $x^\prime$ be the point of incidence and suppose that $w^\prime$ is the new direction of the particle upon hitting the boundary. Then $\phi(x, w) = (x^\prime, w^\prime)$. The Billiard map can also be described in terms of the billiard flow. That is, suppose that $\varphi_t : S^\ast D \to S^\ast D$ is the billiard flow. The billiard flow $\varphi_t$ solves the equation $$ \partial_t\varphi_t(x, \omega) = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\varphi_t(x, w). $$ Then $$ \phi(x, w) = \varphi_{\tau(x,w)}(x, w) $$ where $$ \tau(x, w) = \inf\{t > 0: \varphi_t(x, w) \in M\}. $$

Let $p$ be a given function that is invariant with respect to the billiard map. i.e, $$p:M \to \mathbb{C}$$ is such that $$ p\circ \phi = p \quad \text{on } M. $$ Supposed that the function $p$ is defined on all of $\mathbb{R}^2\times \mathbb{R}^2$. By quantizing $p$, we obtain a pseudo-differential operator $P_h = p(x, hD)$. That is, $$ P_hu(x) = \frac{1}{\left(2\pi h\right)^2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} e^{\frac{i}{h}\langle x-y, \xi\rangle} p(x, \xi) u(y)\,\mathrm{d}{y}\mathrm{d}{\xi} $$ for every sufficiently nice function $u$.

I have shown that $P_h$ shares a complete (in $L^2(D)$) collection of eigenfuntions with the Laplacian $\Delta$. In particular, $P_h$ commutes with the Laplacian $\Delta$.

Is there an intuitive reason why the operator I obtained commutes with the Laplacian? Can anyone provide information regarding the significance (or applications) of such an operator?