Stability of a class of matrices with applications to absorbing boundary conditions for Schrödinger-type equations.
A class of tridiagonal complex matrices \(M_{N+1}(h)\) is investigated. The structure of these matrices is such that the coefficients depend on some positive parameter \(h>0\) and \(N=L/h\) for some positive constant \(L\). First, the stability of a class of matrices without the dependence on \(h\) under some conditions on the coefficients is proved. Then also for the original type of matrices under some conditions on the coefficients is proved that all the eigenvalues have negative real part. Practical applications of this theoretical result presented for numerical solving a Schrödinger type partial differential equation with local absorbing boundary condition of order ABC(1,1) that are developed in a previous paper of the author [ibid. 17, No.~2, 181--188 (2004; reviewed above)]. The numerical problem is derived by using a discretization by finite differences and also a spatial discretization for the boundary condition ABC(1,1) is derived. Than the stability for the matrix of this practical problem, using previous results presented before, is proved.
scientific article; zbMATH DE number 2138573