A previously defined analytic technique of constructing matrix elementsfrom the Bernstein-polynomials (B-poly) has been applied toSchr¨odinger equation. This method after solving generalized eigenvalueproblem yields very accurate eigenenergies and eigenvectors. The numericaleigenvectors and eigenvalues obtained from this process agreewell with exact results of the hydrogen-like systems. Furthermore, accuracyof the numerical spectrum of hydrogen equation depends on thenumber of B-polys being used to construct the analytical matrix elements.Validity of eigenvalues and quality of the constructed wavefunctionsis verified by evaluating the Thomas-Reiche-Kuhn (TRK) sumrules. Excellent numerical agreement is seen with exact results of hydrogenatom.