Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

CivilComp Proceedings
ISSN 17593433 CCP: 99
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping
Paper 53
The Dispersion Function of a Quasi TwoDimensional Periodic Sandwich using a DirichlettoNeumann Map N.T. Bagraev^{1}, G.J. Martin^{2}, B.S. Pavlov^{2,3}, A.M. Yafyasov^{3}, L.I. Goncharov^{3} and A.V. Zubkova^{3}
^{1}Ioffe PhysicalTechnical 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 TwoDimensional Periodic Sandwich using a DirichlettoNeumann Map", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 53, 2012. doi:10.4203/ccp.99.53
Keywords: periodic structures, DirichlettoNeumann map, fitted zerorange model.
Summary
Bloch waves with real quasimomentum play a role in the eigenfunctions of the continuous spectrum of the periodic lattice. In the case of a onedimensional periodic lattice they are obtained as linear combinations of standard solutions of the Cauchy problem for the corresponding onedimensional 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 illposed. Vice versa, the approach based on the boundary problem and the DirichlettoNeumann map of the period is efficient and and even permits extension to periodic sandwich structures. To make the approach efficient for low energy, the quasiperiodic 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 finitedimensional contact spaces. Moreover, the DirichlettoNeumann 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 DNmap permits the calculation of the approximate dispersion function. On another hand, the approximate DNmap 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 onebody 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.
purchase the fulltext of this paper (price £20)
go to the previous paper 
