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Galerkin analysis for Schr\"{o}dinger equation by wavelets
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Dao-Qing Dai, Bin Han and Rong-Qing Jia

## Abstract

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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