ON THE ABSOLUTELY CONTINUOUS SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR ACTING ON p-FORMS FOR A CLASS OF WARPED PRODUCT METRICS

ON THE SPECTRUM OF THE LAPLACE-BELTRAMI OPERATOR FOR p-FORMS FOR A CLASS OF WARPED PRODUCT METRICS

Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating Metrics

We consider a family of manifolds with a class of degenerating warped product metrics gǫ = ρ(ǫ, t) dt + ρ(ǫ, t)2bds2M , with M compact, ρ homogeneous degree one, a ≤ −1 and b > 0. We study the Laplace operator acting on L differential p-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric g0.

An Application of Scattering Theory to the Spectrum of the Laplace-beltrami Operator

Applying a theorem due to Belopol’ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on R endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).

Warped product and quasi-Einstein metrics

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

Absolutely Continuous Spectrum of One Random Elliptic Operator

In dimension d ≥ 5, we consider the differential operator (1.1) H0 = −∆+ τζ(x)|x|(−∆θ), ε > 0, τ > 0, where ∆θ is the Laplace-Beltrami operator on the unit sphere S = {x ∈ R : |x| = 1} and ζ is the characteristic function of the complement to the unit ball {x ∈ R : |x| ≤ 1}. The standard argument with separation of variables allows one to define this operator as the orthogonal sum of one-dimens...