Analysis on Superlattices using Spectral Analysis of Pauli Operator

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Energy filters based on superlattices are a kind of semi-conductor devices using quantum effects. In our study, we investigate this superlattice structure as a mathematical object with considering requirements from the point of industrial view, that is, we want to decide the potential to gain the desired transmission probability. This is an inverse scattering problem on one-dimensional Schroedinger operators. We obtained the following results with the aid of M. Suzuki, H. Sanada and K Asakura.1 We show that Gaussian superlattice modulated layer thickness and one with delta interaction potential form a band structure, which consists of real pass bands, quasi pass bands and stop bands. We propose a method to estimate the real pass bands using image parameters.2 We propose a new kind of effective potential of modulated superlattices. This new potential make us to explain the reason why quasi pass bands appear, that is, the resonances for the double wall structure of the new potential make them.3 We give a sufficient condition for a one-dimensional Schroedinger operator with point delta interactions, which contain m points of interaction with negative intensities, to have at least m negative eigenvalues. This model contains superlattices as special cases.4 We obtain a new functional form of the Green's function of discrete Laplacian L on Z. The new form is a composition of elementary functions (sin, arccos and exponential functions). Using this, we can investigate the spectral properties of L more easily.