The estimates of periodic potentials in terms of effective masses

LetG _n=(A _n ⁻ , A _n ⁺ ),n≧1, denote the gaps,M _n ^± be the effective masses and Σn=A _n−1 ⁺ ,A _n ^- ],A ₀ ⁺ =0, be the spectral bands of the Hill operatorT=−d ²/dx ²+V(x) inL ² (R), whereV is a 1-periodic real potential fromL ²(0,1). Let the length gapL _n=|G_n|, h_n be the height of the corresponding slit on the quasimomentum domain and Δ_n=π²(2n−1)−∣Σ_n∣>0 be the band reduction. Let ,n≧1, denote the gap length for the operator . Introduce the sequencesL={L_n}, h={h_n}, l={l_n}, Δ={Δ_n},M ^±={M _n ^± } and the norms ,m≧0. The following results are obtained: i) The estimates of‖V‖, ‖L‖, ‖h‖ ₁, ‖l‖₁, ‖δ‖ in terms of ‖M^±‖₂, ii) identities for the Dirichlet integral of quasimomentum and integral of potentials and so on, iii) the generation of i), ii) for more general potentials. The research described in this publication was made possible in part by grant from the Russian Fund of Fundamental Research and INTAS.