Invariance of the microsupport of some evolution problems.

This paper is devoted to prove the invariance of the microsupport under the influence of the evolution of the solution in L2(ℝn)$$ {L}^2\left({\mathrm{\mathbb{R}}}^n\right) $$ of the following Cauchy problem: 1(hDt+P)u=0,$$ \left(h{D}_t+P\right)u=0, $$ 2u(0,x)=u0(x),$$ u\left(0,x\right)={u}_0(x), $$where h$$ h $$ is a semiclassical parameter, Dt=1i∂∂t$$ {D}_t=\frac{1}{i}\frac{\partial }{\partial_t} $$ and P=Opth(p)$$ P=O{p}_t^h(p) $$ (t∈[0,1]$$ t\in \left[0,1\right] $$) is a pseudo‐differential operator of symbol p(x,ξ,h)=p0(x,ξ)+ε(h)r(x,ξ,h)∈S2nhol(1),$$ p\left(x,\xi, h\right)={p}_0\left(x,\xi \right)+\varepsilon (h)r\left(x,\xi, h\right)\in {S}_{2n}^{hol}(1), $$where ε(h)→0$$ \varepsilon (h)\to 0 $$ as h→0$$ h\to 0 $$ and p0(x,ξ)$$ {p}_0\left(x,\xi \right) $$ is real valued, that is, p0(x,ξ)∈ℝfor all(x,ξ)∈ℝ2n.$$ {p}_0\left(x,\xi \right)\in \mathrm{\mathbb{R}}\kern0.30em \mathrm{for}\ \mathrm{all}\kern0.1em \left(x,\xi \right)\in {\mathrm{\mathbb{R}}}^{2n}. $$ [ABSTRACT FROM AUTHOR]