Gaps in the Dispersion Relation of the One-dimensional Schrödinger Operator with Periodic Potentials

Download or read book Gaps in the Dispersion Relation of the One-dimensional Schrödinger Operator with Periodic Potentials written by Thomas Z. Dean and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The self-adjoint Schrödinger operator is the difference of a kinetic (Laplacian operator)and potential energy (multiplication operator). The study of this operator continues to attract the interest of many mathematicians and physicists. A commonly used mathematical approach to understand quantum mechanics is through the use of spectral and perturbation theory of the Schrödinger operator. By understanding the spectrum of the Schrödinger operator, we can understand the allowed energy states of a quantum system corresponding to a specific potential. The choice of potential dictates the behavior of the spectrum of the Schrödinger operator which in return provides insight into the behavior of the corresponding quantum system. We study periodic potentials for the Schrödinger operator because of its relation to the phenomena of Anderson localization and semi-conductor theory. A new algorithm is developed to numerically approximate the spectrum of one-dimensional periodic Schrödinger operators. From this, the behavior of spectral gaps are understood when parameters of the potential are changed (e.g. period and amplitude).Moreover, the convergence properties and the behavior of the spectrum as continuous periodic potentials are approximated by their Fourier modes are studied. The behavior of the first spectral gap for such convergences are demonstrated. These results show that the first spectral gap is well-behaved in the strong and norm resolvent convergence.