Galerkin analysis for Schr\"{o}dinger equation by wavelets

Dao-Qing Dai, Bin Han and Rong-Qing Jia


We consider perturbed Schr\"{o}dinger equation, which is an elliptic operator with unbounded coefficients. We use wavelets constructed from Hermite functions to approximate the solutions by the Galerkin method. This wavelet characterizes the space generated by the Schr\"{o}dinger 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.

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