Vol. 37, No. 1, 1971

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On spectral theory and scattering for elliptic operators with singular potentials

Richard William Beals

Vol. 37 (1971), No. 1, 7–20

General conditions have been found which imply that the perturbation A + q of an elliptic differential operator A by a singular potential term q(x) has a closed extension B in L2(Rn) having the same essential spectrum as A. The purpose of this paper is to sharpen the known results slightly and to estimate the characteristic numbers of the operator (A + λ)p (B + λ)p. Under an appropriate assumption on q(x), this operator is shown to be of trace class for large p. In the self-adjoint case it follows then from results of Kato that wave operators for the pair (A,B) exist and that the absolutely continuous parts of these operators are unitarily equivalent.

Mathematical Subject Classification 2000
Primary: 35P25
Received: 4 August 1969
Revised: 11 September 1970
Published: 1 April 1971
Richard William Beals