Galerkin analysis for Schrödinger equation by wavelets

We consider perturbed Schrödinger equation, which is an elliptic operator with unbounded coefficients. We use wavelets adapted to Schrödinger operator to deal with problems on 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 preconditioned 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 L2 topology. 02.30.G: Special functions 02.30.L: Sequences, series and summability 02.30.Q: Integral transforms and operational calculus