THRESHOLD APPROXIMATIONS FOR A FACTORIZED SELFADJOINT OPERATOR FAMILY WITH THE FIRST AND SECOND CORRECTORS TAKEN INTO ACCOUNT

摘要

In a Hilbert space Sj, a family of operators A(t) admitting a factorization of the form A(t) = Х(t)~*Х(t), where X(t) = Xo +tX_1, t∈ R, is considered. It is assumed that the point λo = 0 is an isolated eigenvalue of finite multiplicity for A(0). Let F(t) be the spectral projection of A(t) for the interval [0, δ] (where δ is sufficiently small). For small |t|, approximations in the operator norm in fj are obtained for the projection F(t) with an error of О(|t|~3) and for the operator A(t)F(t) with an error of О(|t|~5) (the threshold approximations). By using these results, approximation in the operator norm in fj are constructed for the operator exponential exр(-A(t)T) for large T > O with an error of О(T~(-3/2)). For the resolvent (A(t) + ∈~2I)~(_1) multiplied by a suitable "smoothing' factor, approximation in the operator norm in for small ε > 0 with an error of 0(ε) is obtained. All approximations are given in terms of the spectral characteristics of A(t) near the bottom of the spectrum. In these approximations, the first and the second correctors are taken into account. The results are aimed at applications to homogenization problems for periodic differential operators in the small period limit.