From the 1 of 7 linked papers with an AI index.
7 papers
Log-Concavity and Level-Set Horoconvexity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane
Xianzhe Dai, John M. Ennis, Xuan Hien Nguyen +1
The paper proves that the first Dirichlet eigenfunction on any bounded smooth horoconvex domain in the hyperbolic plane is log‑concave, i.e., the Hessian of its negative logarithm…
A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains
Xianzhe Dai, John Ennis, Xuan Hien Nguyen +1
We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domai…
The classical Weyl law for Schrödinger operators on complete Riemannian manifolds
Maxim Braverman, Xianzhe Dai, Junrong Yan
We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schrödinger operators on complete Riemannian manifolds. In contrast to exist…
Degeneration of Riemann surfaces and small eigenvalues of the Laplacian
Xianzhe Dai, Ken-Ichi Yoshikawa
For a one-parameter degeneration of compact Riemann surfaces endowed with the Kähler metric induced from the Kähler metric on the total space of the family, we determine the exac…
Adiabatic Limit and Analytic Torsion of Vector Bundles
Xianzhe Dai, Debin Liu
For a vector bundle over a closed manifold with even and odd, we equip the metric with an adiabatic parameter, and prove that the index of is the same a…
Weyl Law for Schrödinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem
Xianzhe Dai, Junrong Yan
Building on our earlier work on heat kernel asymptotics for Schrödinger-type operators on noncompact manifolds, we establish both the classical and semiclassical Weyl laws for Sch…