I hope that is an applicable query for this discussion board. If not, I apologize. Earlier than stating my query (which can be discovered on the finish of this submit), I’ll try to offer ample context.

I’m finding out eigenfunctions of the Laplacian (each Dirichlet and Neumann circumstances) on a specified bounded area $Dsubseteq mathbb{R}^2$ which has piece-wise easy boundary. Subsequent, think about the billiard map

$$phi : M to M,$$

the place $M$ is the subset of the unit cotangent bundle $S^ast(partial D)$ containing solely the inward pointing instructions. The billiard map fashions the behaviour of a free particle within the area $D$. Suppose we’re base level $xin partial D$ and a unit path vector $w$ pointing inwards. Then a free particle beginning at x and travelling in path $w$ will finally hit the boundary $partial D$. Let $x^prime$ be the purpose of incidence and suppose that $w^prime$ is the brand new path of the particle upon hitting the boundary. Then $phi(x, w) = (x^prime, w^prime)$. The Billiard map will also be described by way of the billiard circulate. That’s, suppose that $varphi_t : S^ast D to S^ast D$ is the billiard circulate. The billiard circulate $varphi_t$ solves the equation

$$

partial_tvarphi_t(x, omega) = start{pmatrix}

0 & 1

0 & 0

finish{pmatrix}varphi_t(x, w).

$$

Then

$$

phi(x, w) = varphi_{tau(x,w)}(x, w)

$$

the place

$$

tau(x, w) = inf{t > 0: varphi_t(x, w) in M}.

$$

Let $p$ be a given operate that’s invariant with respect to the billiard map. i.e, $$p:M to mathbb{C}$$ is such that

$$

pcirc phi = p quad textual content{on } M.

$$

We notice that the operate $p$ is outlined on all of $mathbb{R}^2times mathbb{R}^2$. By quantizing $p$, we receive a pseudo-differential operator $P_h = p(x, hD)$. That’s,

$$

P_hu(x) = frac{1}{left(2pi hright)^2}int_{mathbb{R}^2}int_{mathbb{R}^2} e^{frac{i}{h}langle x-y, xirangle} p(x, xi) u(y),mathrm{d}{y}mathrm{d}{xi}

$$

for each sufficiently good operate $u$.

I’ve proven that $P_h$ shares a whole (in $L^2(D)$) assortment of eigenfuntions with the Laplacian $Delta$. Specifically, $P_h$ commutes with the Laplacian $Delta$.

Is there an intuitive purpose why the operator I obtained commutes with the Laplacian? Can anybody present data concerning the importance (or purposes) of such an operator?