A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

In this work, a new approximation scheme based on the evaluation of the pointwise expectation of the Hamiltonian (H) via a conveniently chosen basis set is proposed. This scheme does not necessitate integration; however, physical and mathematical considerations in choosing the basis set are considerably important when very precise and rapidly convergent results are desired. In this method, the best linear combination of “well‐selected” basis functions are sought in a way such that H ψ / ψ is flat in the neighborhood of a conveniently chosen point in the domain of H. This yields an algebraic eigenvalue problem. Some concrete applications that have already been realized confirm the efficiency of this approach.