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We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H = −d 2 /dx 2 + V in L 2 (R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of… (More)

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrödinger operator) H = −d 2 /dx 2 + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the… (More)

- Vadim Tkachenko
- 2008

We prove an explicit formula for the spectral expansions in L 2 (R) generated by selfadjoint differential operators (−1) n d 2n dx 2n + n−1 j=0 d j dx j pj(x) d j dx j , pj(x + π) = pj(x), x ∈ R. It is well known [1], see also [2], that for every Hill operator H = − d 2 dx 2 + q(x), q(x) = q(x + π), x ∈ R (1.1) with a real-valued potential function q(x)… (More)

The mathematical model of volume optical interconnection elements that are formed within the LiNbO<sub>3</sub> crystal is proposed in the paper. The simulations using this model were conducted. The possibility to reconfigure the structure of optical interconnections was demonstrated.

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