Threshold approximations for the exponential of a factorized operator family with correctors taken into account

Abstract

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.