We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial {}_{t}u=\mathrm{△}u+\frac{c}{\left|x^2\right|}u(0<c<\frac{{\left(n-2\right)}^2}{4};n\ge 3)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu=-\mathrm{△}u-\frac{c}{{\left|x\right|}^2}u$ with the opposite of the weighted Laplacian ${\mathrm{△}}_{\lambda }v=\frac{1}{{\left|x\right|}^{\lambda }}\text{div}\left({\left|x\right|}^{\lambda }\nabla v\right)$ when $\lambda =2-n+2\sqrt{c_0-c}$.

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the...

We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a $$𝔤{𝔩}_n$$ partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.