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We derive simple formulas for the matrix elements of the resolvent operator (also known as, the Green’s function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are orthogonal or not. The formulas are written in terms of the eigenvalues and normalized eigenvectors of the matrix representation of the associated operator on the said basis. To reduce the computational cost, we also present a version of the same formulas using only matrix eigenvalues without the need for the sumptuous calculation of normalized eigenvectors. A byproduct of our findings is an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues. We give a physical application of how useful these results can be. As an illustration, we use our findings to locate the resonances of a quantum mechanical system, obtain its bound states energies, and plot its energy density of states.
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Matrix Representation of the Resolvent Operator in Square-integrable Basis and Physical Application
How to cite this paper: A. D. Alhaidari. (2024) Matrix Representation of the Resolvent Operator in Square-integrable Basis and Physical Application. Journal of Applied Mathematics and Computation, 8(4), 296-307.
DOI: http://dx.doi.org/10.26855/jamc.2024.12.003