The paper deals with the Sturm-Liouville operator Ly = −y ′′ + q(x)y,x ∈ [0, 1], generated in the space L 2 = L 2 [0, 1] by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to Sobolev space W p 1 [0, 1] with some integer p ≥ 0 and satisfy the conditionsLet the functions Q and S be defined by the equalities Q(x) = x 0 q(t) dt, S(x) = Q 2 (x) and let q n , Q n , S n be the Fourier coefficients of q, Q, S with respect to the trigonometric system {e 2πinx } ∞ −∞ . Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2 . Then the system of eigen and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2 [0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q 2n − S 2n + Q 0 Q 2n ≍ q −2n − S −2n + 2Q 0 Q −2n , n > 1, holds.
We have experimentally demonstrated the interferometric complementarity, which relates the distinguishability D quantifying the amount of which-way (WW) information to the fringe visibility V characterizing the wave feature of a quantum entity, in a bulk ensemble by nuclear magnetic resonance (NMR) techniques. We are primarily concerned about the intermediate cases: partial fringe visibility and incomplete WW information. We propose a quantitative measure of D by an alternative geometric strategy and investigate the relation between D and entanglement. By measuring D and V independently, it turns out that the duality relation D 2 + V 2 = 1 holds for pure quantum states of the markers.
moreover, since /1 and u are equivalent on S, we see that iii) /zt~ and vt~ are equivalent on S + tz for all t 6 R. It follows from Lemma 3 that for an H-quasi-invariant measure /z there exists a t0 6 R such that ~(Sn (S + toz)) >0.Then conditions i), ii), and iii) imply that the measures # and u are singular on S N (S + t0x). This contradicts the fact that /~ and u are equivalent on S. [] The author expresses his deep gratitude to D. Kh. Mushtari for permanent attention to the work.
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