Complex powers of the Schr?dinger operator with singular potential

摘要

We consider in R~n (n> 3) the Schr?dinger operator H(x, D)= --△ + V(x) and suppose that potential V is smooth enough outside S C R~n, where {formula} and Sk are smooth surfaces in R~n of dimension n--mk, {formula}. We suppose that for some r, {formula}, for all multiindices a with |α| ≤ n. As an example for such kind of operators we can take potential of many particle systems. It is proved that imaginary powers of these operators are bounded in L_p(R~n). This fact and E.Steyn interpolation theorem gives us possibility to estimate of real fractional powers of the Schr?dinger operator.