Semiclassical scar on tori in high dimension

We show that the eigenfunctions of the self-adjoint elliptic $h-$differential operator $P_h(t)$ exhibits semiclassical scar phenomena on the $d-$dimensional torus, under the $\sigma$-Bruno-Rüssmann condition, instead of the Diophantine one. Its equivalence is described as: for almost all perturbed Hamiltonian’s KAM Lagrangian tori ${\mathrm{\Lambda }}_{\omega }$, there exists a semiclassical measure with positive mass on ${\mathrm{\Lambda }}_{\omega }$. The premise is that we can obatain a family of quasimodes for the $h-$differential operator $P_h(t)$ in the semiclassical limit as $h\to 0$, under the $\sigma$-Bruno-Rüssmann condition.