Spectral radius, index estimates for Schrodinger operators and geometric applications

TL;DR

This paper introduces new techniques to analyze the zeros of differential equations linked to geometric functions, applying these to estimate spectral properties of Schrödinger operators and Laplacians on Riemannian manifolds, with implications for minimal hypersurfaces and the Yamabe problem.

Contribution

It presents a novel method for estimating zero distances in differential equations and applies it to derive spectral and geometric bounds on Riemannian manifolds.

Findings

New zero-distance estimation technique for differential equations

Bounds on spectral radius growth of Laplacian on manifolds

Applications to minimal hypersurfaces and Yamabe problem

Abstract

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation $(vz')'+Avz = 0$, where $A,v$ are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schr\"odinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.