Analysis And Efficient Solution Of Stationary Schrödinger Equation Governing Electronic States Of Quantum Dots And Rings In Magnetic Field

2012 ◽  
Vol 11 (5) ◽  
pp. 1591-1617 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Magnetic Field ◽  
Eigenvalue Problems ◽  
Rotational Symmetry ◽  
Electronic States ◽  
Nonlinear Problems ◽  
Projection Methods ◽  
Computational Time ◽  
Arnoldi Method ◽  
Nonlinear Eigenvalue Problems ◽  
The One

AbstractIn this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number -L•i. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

scholarly journals A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Sandile S. Motsa ◽  
Precious Sibanda
Keyword(s):  
Eigenvalue Problems ◽  
Duffing Oscillator ◽  
Nonlinear Problems ◽  
Governing Equations ◽  
Van Der Pol ◽  
Oscillatory Solutions ◽  
Nonlinear Eigenvalue Problems ◽  
Successive Linearisation Method ◽  
Oscillator Equations ◽  
Nonlinear Eigenvalue

We present a novel application of the successive linearisation method to the classical Van der Pol and Duffing oscillator equations. By recasting the governing equations as nonlinear eigenvalue problems we obtain accurate values of the frequency and amplitude. We demonstrate that the proposed method can be used to obtain the limit cycle and bifurcation diagrams of the governing equations. Comparison with exact and other results in the literature shows that the method is accurate and effective in finding solutions of nonlinear equations with oscillatory solutions, nonlinear eigenvalue problems, and other nonlinear problems with bifurcations.

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

2008 ◽  
Vol 13 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Solid Structure ◽  
Minmax Characterization ◽  
Nonlinear Eigenvalue

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

The Energy Band Structure of Polyacene with Soliton Excitation

1997 ◽  
Vol 11 (11) ◽  
pp. 477-483 ◽  
Author(s):  
Z. J. Li ◽  
H. B. Xu ◽  
K. L. Yao
Keyword(s):  
Band Structure ◽  
Charge Density ◽  
Energy Band ◽  
Energy Gap ◽  
Electronic States ◽  
Energy Band Structure ◽  
Energy Spectra ◽  
Energy Band Gap ◽  
The One ◽  
Soliton Excitation

Starting from the extensional Su–Schrieffer–Heeger model taking into account the effects of interchain coupling, we have studied the energy spectra and electronic states of soliton excitation in polyacene. The dimerized displacement u0 is found to be similar to the case of trans-polyacetylene, and equals to 0.04 Å. The energy-band gap is 0.38 eV, in agreement with the results derived by other authors. Two new bound electronic states have been found in the conduction band and in the valence band, which is different from the one of trans-polyacetylene. There exists two degenerate soliton states in the center of energy gap. Furthermore, the distribution of charge density and spin density have been discussed in detail.

Sign in / Sign up

Export Citation Format

Share Document