A conjecture for some partial differential operators onL 2(R n)
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OnX =L 2(R n), letQ = (Q 1,Q 2,…,Q n) andP = (P 1,P 2, …,P n) be the operators given by (Q jf) (x) =x jf(x),P j = - i∂/∂x j. For anyC ∞ functionh:R n →R putH 0 =h(P) andH =H 0 + (1 +Q 2)-δ, where δ > 1/2. By the method of scattering theory we prove thatH ac, the absolutely continuous part ofH is unitarily equivalent toH 0 when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.
KeywordsPartial differential operators scattering theory Enss’ method
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