Numerical instability in Rayleigh-Schrodinger quantum mechanics

The most physically interesting systems are not exactly solvable in quantum mechanics. For one-dimensional bound systems without exact solutions, we a nalytically and numerically find that the Rayleigh-Schrodinger perturbed se ries sensitively depends on an unsolvable integration, which leads to numer ical instability in quantum mechanics. By using an exact formal solution of the non-homogeneous Schrodinger equation, we demonstrate the existence of analytically bound states and propose a simple scheme to truncate infinity so that the instability difficulty is avoided.