Galerkin analysis for Schrödinger equation by wavelets. (English) Zbl 1070.65142
Summary: We consider the perturbed Schrödinger equation, which is an elliptic operator with unbounded coefficients. We use wavelets adapted to the Schrödinger operator to deal with problems on the unbounded domain. The wavelets are constructed from Hermite functions, which characterizes the space generated by the Schrödinger operator. We show that the Galerkin matrix can be pre-conditioned by a diagonal matrix so that its condition number is uniformly bounded. Moreover, we introduce a periodic pseudo-differential operator and show that its discrete Galerkin matrix under periodic wavelet system is equal to the Galerkin matrix for the equation with unbounded coefficients under the Hermite system. The convergence is proved in the \(L^2\) topology.
MSC:
| 65T60 | Numerical methods for wavelets |
| 34B24 | Sturm-Liouville theory |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
References:
| [1] | DOI: 10.1002/cpa.3160440202 · Zbl 0722.65022 · doi:10.1002/cpa.3160440202 |
| [2] | Dahmen W., Math. Z. 215 pp 583– (1994) · Zbl 0794.65082 · doi:10.1007/BF02571732 |
| [3] | Dahmen W., Adv. Comput. Math. 1 pp 259– (1993) · Zbl 0826.65093 · doi:10.1007/BF02072014 |
| [4] | Dahmen W., J. Comput. Appl. Math. 128 pp 133– (2001) · Zbl 0974.65101 · doi:10.1016/S0377-0427(00)00511-2 |
| [5] | Dai D. Q., J. Math. Phys. 41 pp 3086– (2000) · Zbl 0973.42027 · doi:10.1063/1.533293 |
| [6] | Frazier M., J. Math. Anal. Appl. 261 pp 665– (2001) · Zbl 1002.65084 · doi:10.1006/jmaa.2001.7567 |
| [7] | Funaro D., Math. Comput. 57 pp 597– (1990) · doi:10.1090/S0025-5718-1991-1094949-X |
| [8] | Guo B., Math. Comput. 68 pp 1067– (1999) · Zbl 0918.65069 · doi:10.1090/S0025-5718-99-01059-5 |
| [9] | Jaffard S., SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 29 pp 965– (1992) · Zbl 0761.65083 · doi:10.1137/0729059 |
| [10] | Krall A. M., Trans. Am. Math. Soc. 344 pp 155– (1994) · doi:10.1090/S0002-9947-1994-1242783-9 |
| [11] | Lewalle J., J. Math. Phys. 39 pp 4119– (1998) · Zbl 0933.35163 · doi:10.1063/1.532487 |
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.
