Schrödinger Equation with Periodic potentials

The Schrödinger equation ∂2xΨ(x) = [U(x) − E]Ψ(x) with the potential U(x) = 2ρ1 cos x + 2ρ2 cos(γx) is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sen-sitive to the value of the parameter γ. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter γ. I Acknowledgements I would like to acknowledge the people who have helped me along the way. First, I am heartily thankful to my supervisor, Professor A. Vourdas, who encouragement, guidance and support from the initial to the final level en-abled me to develop an understanding of the subject. I am indebted to Garyounis University and Libyan Cultural Affairs for all the financial support. I wish to thank the School of Computing, Informatics and Media at University of Bradford for helped me during my study. I am especially grateful for my parents, for their support and encourage me to do the best in the life. Lastly, and most importantly, I would like to dedicate this thesis to the persons, who have made this work successful, and all the acknowledge words not enough to them, my husband Salah Elhassi and my sons, Saleh, Sanad and Ibrahim. I