Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-02T21:37:36.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Schrödinger-Poisson systems in dimension d ≦ 3: The whole-space case

Published online by Cambridge University Press:  14 November 2011

F. Nier
F. Nier
Affiliation:
CMAP-Ecole Poly technique, URA-CNRS 756, F-91128 Palaiseau Cedex, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

After a study made for bounded domains and in the periodic case, we investigate the variational formulation of Schrödinger-Poisson systems set on the whole space ℝd, d ≦ 3. This variational formulation leads to a uniqueness result, while the existence of a solution is proved only for ‘small data’ because of the lack of coerciveness. The end of this paper briefly presents the extension of this formalism to a physically relevant problem where the potential is periodic in one direction.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 6 , 1993 , pp. 1179 - 1201
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Agmon, S.. Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa 2 (1975), 151–218.Google Scholar
3Arnold, A., Markowich, P. A. and Mauser, N.. The one-dimensional periodic Bloch-Poisson equation. M3AS 1 (1991), 83–112.Google Scholar
4Beaumont, S. P. and Sotomayor-Torres, C. M. (eds). Science and Engineering of One and Zero-Dimensional Semiconductors (New York: Plenum, 1990).Google Scholar
5Brezis, H.. Analyse Fonctionnelle: theorie et applications (Paris: Masson, 1983).Google Scholar
6Caussignac, P., Zimmermann, B. and Ferro, R.. Finite element approximation of electrostatic potential in one-dimensional multilayer structures with quantized electronic charge. Computing 45 (1990), 25.CrossRefGoogle Scholar
7Chazarain, J. and Piriou, A.. Introduction a la Theorie des Equations aux Derivees Partielles Lineaires (Paris: Gauthiers-Villars, 1981).Google Scholar
8Cycon, H. L., Froese, R., Kirsch, W. and Simon, B.. Schrodinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics (Berlin: Springer, 1987).Google Scholar
9Gerard, C.. Sharp propagation estimates for N-particles systems (Preprint—CMAT Ecole Polytechnique—91128 Palaiseau Cedex, France, 1990).Google Scholar
10Helffer, B. and Sjöstrand, J.. Equation de Harper. Lecture Notes in Physics 345 (1989), 118–197.CrossRefGoogle Scholar
11Kerkhoven, T., Galick, A. T., Ravaioli, V., Arends, J. H. and Saad, Y.. Efficient numerical simulation of electron states in quantum wires. J. Appl. Phys. 68 (1990), 3461–3469.CrossRefGoogle Scholar
12Kerkhoven, T., Raschke, M. W. and Ravaioli, V.. Self-consistent simulation of quantum wires in periodic heterojunction structures (manuscript).Google Scholar
13Kajima, K., Mitsunaga, K. and Kyuma, K.. Calculation of two-dimensional quantum-confined structures using the finite element method. Appl. Phys. Lett. 55 (1989), 882–884.Google Scholar
14Lions, J. L. and Magenes, E.. Problemes aux Limites non homogenes et Applications (Paris'. Dunod, 1968).Google Scholar
15Markowich, P. A.. Boltzmann distributed quantum steady-states and their classical limit. Forum Math, (to appear).Google Scholar
16Nier, F.. A stationary Schrodinger-Poisson system arising from the modelling of electronic devices. Forum Math 2 (1990), 489–510.CrossRefGoogle Scholar
17Nier, F.. A variational formulation of Schrödinger-Poisson systems in dimension d≤3 (Technical Report #166, Purdue University, West Lafayette, IN 47907); Comm. Partial Differential Equations (submitted).Google Scholar
18Nirenberg, L.. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13 (1959), 116–162.Google Scholar
19Reed, M. and Simon, B.. Methods of Modern Mathematical Physics (New York: Academic Press, 1975).Google Scholar
20Schubin, M. A.. The spectral theory and the index of elliptic operators with almost periodic coefficients. Russian Math. Surveys 34.2 (1979), 109–157.Google Scholar
21Simon, B.. Trace Ideals and Applications (Cambridge: Cambridge University Press, 1978).Google Scholar
22Simon, B.. Functional Integration and Quantum Physics (New York: Academic Press, Cambridge: 1979).Google Scholar
23Vinter, B.. Subbands and charge control in a two-dimensional electron gas field effect transistor. Appl. Phys. Let. 44 (1987), 307.Google Scholar