Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

Abstract

In this paper, we consider the eigen-solutions of −∆u+V u = λu, where ∆ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions as r goes to infinity based on the asymptotical behaviors of ∆r and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies K rad (r) = −1 + o(1) r .

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

“…They go back to Pinsky [25], with a later milestone by Donnelly [4]. Some recent results on the absence of eigenvalues can be found in [19,21].…”

Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

“…Most of the past work has been focused on proofs of the purity of absolutely continuous spectrum, guaranteed by the asymptotic curvature conditions, going back to [6,26]. Several extensions of purity results have also appeared recently [10,11,23,24]. Lately, some attention has turned to the opposite phenomenon.…”

“…Note that compact perturbations of constant curvature can only lead to eigenvalues below the essential spectrum, so embedding questions are naturally tied to the rate of decay. Sharp decay thresholds have been established for existence of metrics with an embedded eigenvalue [20] (see also [23] for a simple proof of sharpness) and with an embedded arbitrary countable (in particular, dense) set [13] (also for the flat, i.e. K 0 = 0, case).…”

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds II: Positivity of the Initial Energy

“…Let r be the distance function from ∂U defined on the end M −U . This is the same setting as in [18,20,23].…”

“…His basic idea is (four steps): construct energy function for eigen-equation; set up the positivity of initial energy; prove the monotonicity of energy function with respect to r; obtain the growth of eigen-solution. In our previous paper [23], we have already used such scheme to set up the growth of eigen-solutions by a universal construction of energy functions. As an application, we showed the absence of eigenvalues in certain circumstances, which improved Kumura's sharp results [18] mentioned before.…”