Global Recursion Polynomial Expansions of the Green’s Function and Time Evolution Operator for the Schrödinger Equation with Absorbing Boundary Conditions
We review and revise some recently developed iterative numerical techniques of solving the Time-Independent Wave Packet Schrödinger Equation (E—Ĥ)Ψ = X, which are especially suitable for calculations associated with a large many body scattering problem, requiring large basis sets (grids). The methods we consider take advantage of the particular way the wave equation depends on the energy E. Namely, when global polynomial expansions of the resolvent operator are used the energy-dependent solution can be generated simultaneously at many energies from essentially a single iterative procedure. A general problem in constructing a well behaved polynomial expansion of a function of operator argument, f(Ĥ), is how to incorporate absorbing boundary conditions (ABC) which would eliminate reflection effects caused by an artificial truncation of an infinite grid where Ĥ is defined. It is shown that this problem can be solved naturally by modifying the auxiliary equations (recursion relations) used to generate the interpolating polynomials; this avoids using non-Hermitian operators. In particular Chebyshev and Newtonian interpolation schemes are considered. While the latter is more general, the former allows one to obtain analytically very simple global recursion polynomial expansions for the ABC Green’s function and ABC time evolution operator.
KeywordsRecursion Relation Schrodinger Equation Polynomial Expansion Resolvent Operator Absorb Boundary Condition
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