The Laplacian Δ for Euclidean space Rn has the following properties: (a) the essential spectrum of -Δ is [0, ∞); (b) Δ has no point spectrum, and (c) Δ has no singular continuous spectnim. If (xi , x2 , . . . , xn.) are the standard global coor- dinates on Rn, then the exhaustion function b(x) = (x~2_1 + x~2_2 + . . . + x~2_n)1/2 sat- isfies (i) |Δb| = 1 for x ≠f 0 and (ii) Hess b≠ = Zg. Here g denotes the Euclidean metrei. Let M be a complete Riemannian manifold that admits a proper exhaustion function b. If (i) and (ii) above are satisfied in a weak or approximate sense, then we would like to show that the Laplacian Δ of M has properties similar to those of the Euclidean Laplacian. This program was started in our earlier paper [6]. Under gencral averaged L2 conditions on |Δ b| and [Δb[-1| , we showed that the es- sential spectrum of -Δ is [0, ∞). More stringent pointwise decay conditions for |Hess b2 - Zg | and [ [Δbl - l [ were needed to eliminate the possibility of a point spectrum for Δ . The singular continuous spectrum was not discussed in [6].
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