Decay estimates of discretized Green's functions for Schrödinger type operators
For researchers in numerical analysis and quantum mechanics, this work provides rigorous decay estimates for discretized Green's functions, but the results are incremental as they extend known decay properties to specific discretizations.
This paper provides decay estimates for discretized Green's functions of Schrödinger type operators, showing that the off-diagonal decay rate is independent of domain size and discretization parameter, and verifies the estimates numerically for one-dimensional operators.
For a sparse non-singular matrix $A$, generally $A^{-1}$ is a dense matrix. However, for a class of matrices, $A^{-1}$ can be a matrix with off-diagonal decay properties, i.e. $\lvert A^{-1}_{ij}\rvert$ decays fast to $0$ with respect to the increase of a properly defined distance between $i$ and $j$. Here we consider the off-diagonal decay properties of discretized Green's functions for Schrödinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schrödinger type operators.