学术报告（戴先哲 2026.7.8）

摘要：The Weyl law for Schrödinger operators on Euclidean spaces is established by Rozenbljum more than fifty years ago. In joint work with Junrong Yan, we prove its generalization to complete manifolds under bounded geometry, volume doubling, and certain oscillation conditions on the potentials. These conditions, while quite natural, rule out certain interesting geometric situations. In this talk, I will present a general criterion for the validity of the Weyl law for Schrödinger operators on complete Riemannian manifolds. We introduce a geometric–analytic invariant that captures the precise balance between the geometry of the manifold, the growth of the potential, and its oscillation scale.

We show that the classical Weyl asymptotic holds under the vanishing of this invariant, and provide explicit examples showing that the Weyl law may fail when this criterion is violated. This illustrates both the nontriviality and the sharpness of our theorem.

This is joint work with Maxim Braverman and Junrong Yan.