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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping
The Dispersion Function of a Quasi Two-Dimensional Periodic Sandwich using a Dirichlet-to-Neumann Map
N.T. Bagraev1, G.J. Martin2, B.S. Pavlov2,3, A.M. Yafyasov3, L.I. Goncharov3 and A.V. Zubkova3
1Ioffe Physical-Technical Institute, Russian Academy of Sciences, St. Petersburg, Russia
N.T. Bagraev, G.J. Martin, B.S. Pavlov, A.M. Yafyasov, L.I. Goncharov, A.V. Zubkova, "The Dispersion Function of a Quasi Two-Dimensional Periodic Sandwich using a Dirichlet-to-Neumann Map", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 53, 2012. doi:10.4203/ccp.99.53
Keywords: periodic structures, Dirichlet-to-Neumann map, fitted zero-range model.
Bloch waves with real quasi-momentum play a role in the eigenfunctions of the continuous spectrum of the periodic lattice. In the case of a one-dimensional periodic lattice they are obtained as linear combinations of standard solutions of the Cauchy problem for the corresponding one-dimensional Schrödinger equation. The Cauchy problem approach to the calculation of the dispersion function fails in the case of the multidimensional Schrödinger equation, because the Cauchy problem for the Schrödinger equation in that case is ill-posed. Vice versa, the approach based on the boundary problem and the Dirichlet-to-Neumann map of the period is efficient and and even permits extension to periodic sandwich structures. To make the approach efficient for low energy, the quasi-periodic matching condition for the wave functions (e.g. Bloch functions) has been subsitiuted on the mutual boundary of the neighbouring periods using an appropriate partial matching on selected contact zones in appropriate finite-dimensional contact spaces. Moreover, the Dirichlet-to-Neumann map, can be substituted for low temperature, using an appropriate rational approximation taking into account only the eigenvalues of the Dirichlet problem for the Schrödinger operator for the period which are situated on the temperature interval centered at the Fermi level. The rational approximation of the DN-map permits the calculation of the approximate dispersion function. On another hand, the approximate DN-map can be related to a model periodic operator which can be interpreted as a fitted solvable model of the original Schrödinger operator. The approximate dispersion function is an exact dispersion function of the fitted solvable model. Hence, based on the fitted model an algorithm which permits the calculation of the approximate dispersion function of the one-body quantum spectral problem on a periodic lattice or sandwich is suggested. The result is presented in the form of an explicit formula, providing the approximate dispersion function for dependence of the shapes of the resonance eigenfunctions of the Dirichlet problem for the period and the characteristics of the selected contact zones and contact spaces. The spectral characteristics of the model can be used as the first step of a convergent analytic perturbation procedure leading to the calculation of the corresponding spectral characteristics of the original perturbed problem.
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