The estimates of periodic potentials in terms of effective masses

Abstract

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$l_n = \sqrt {A_n^ + } - \sqrt {A_n^ - } ,n \geqq 1$,n≧1, denote the gap length for the operator$\sqrt T \geqq 0$. Introduce the sequencesL={L_n}, h={h_n}, l={l_n}, Δ={Δ_n},M ^±={M ^±_n } and the norms$\left\| f \right\|_m^2 = \sum\nolimits_{n > 0} {\left( {2\pi n} \right)^{2m} f_n^2 ,m \geqq 0}$,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.