Existence of an extremal ground state energy of a nanostructured quantum dot. (English) Zbl 1231.35127
Summary: This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 81Q37 | Quantum dots, waveguides, ratchets, etc. |
Keywords:
minimization problems; rearrangement of a function; nonlinear eigenvalue problems; nanostructured quantum dotsReferences:
| [1] | Voss, H.; Werner, B., A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems, Math. Methods Appl. Sci., 4, 415-424 (1982) · Zbl 0489.49029 |
| [2] | Voss, H., A minimax principle for nonlinear eigenvalue problems with applications to a rational spectral problem in fluid-solid vibration, Appl. Math., 48, 607-622 (2003) · Zbl 1099.35076 |
| [3] | Dunford, N.; Schwartz, T., (Linear Operators. Part II. Spectral Theory. Linear Operators. Part II. Spectral Theory, Self Adjoint Operators in Hilbert Space (1988), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York) · Zbl 0635.47002 |
| [4] | Anane, A., Simplicité et isolation de la premiére valeur propre du \(p\)-Laplacien avec poids, C. R. Acad. Sci. Paris Ser. I Math., 305, 725-728 (1987) · Zbl 0633.35061 |
| [5] | Lê, A., Eigenvalue problems for the \(p\)-Laplacian, Nonlinear Anal., 64, 1057-1099 (2006) · Zbl 1208.35015 |
| [6] | Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43, 235-286 (2001) · Zbl 0985.65028 |
| [7] | Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials (1982), Academic Press: Academic Press New York · Zbl 0482.15001 |
| [8] | Lancaster, P., Lambda-Matrices and Vibrating Systems (2002), Dover Publications: Dover Publications Mineola, NY · Zbl 1048.34002 |
| [9] | Przemieniecki, J. S., Theory of Matrix Structural Analysis (1968), MacGraw-Hill: MacGraw-Hill New York · Zbl 0177.53201 |
| [10] | Emamizadeh, B.; Fernandes, R. I., Optimization of the principal eigenvalue of the one-dimensional Schrödinger operator, Electron. J. Differential Equations, 65, 1-11 (2008) · Zbl 1177.34104 |
| [11] | Cuccu, F.; Porru, G., Optimization in eigenvalue problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, 51-58 (2003) · Zbl 1044.35036 |
| [12] | Cuccu, F.; Porru, G., Maximization of the first eigenvalue in problems involving the bi-Laplacian, Nonlinear Anal., 71, e800-e809 (2009) · Zbl 1238.35066 |
| [13] | Emamizadeh, B.; Prajapat, J. V., Symmetry in rearrangement optimization problems, Electron. J. Differential Equations, 149, 1-10 (2009) · Zbl 1193.49003 |
| [14] | Hwang, T. M.; Lin, W. W.; Wang, W. C.; Wang, W., Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys., 196, 208-232 (2004) · Zbl 1053.81018 |
| [15] | Li, Y.; Voskoboynikov, O.; Lee, C. P.; Sze, S. M., Computer simulation of electron energy levels for different shape InAs/GaAs semiconductor quantum dots, Comput. Phys. Commun., 141, 66-72 (2001) · Zbl 1035.81619 |
| [16] | Burton, G. R., Rearrangements of functions, maximization of convex functionals and vortex rings, Math. Ann., 276, 225-253 (1987) · Zbl 0592.35049 |
| [17] | Burton, G. R., Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, 4, 295-319 (1989) · Zbl 0677.49005 |
| [18] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0691.35001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
