Being guided by the problem of bound states in potentials close to their Padé approximants, a new Rayleigh–Schrödinger‐type perturbation theory is developed. The unperturbed system is understood: here in a broader sense: its solutions are not needed, but merely the related nondiagonal unperturbed propagator R
. In particular, all the chain models H0
=band matrices) with arbitrary perturbations are then perturbatively solvable, with R
constructed in terms of auxiliary matrix continued fraction fn
. Alternatively, a ‘‘generalized unperturbed spectrum’’ n
may be required as an input: The algebraically constructed asymptotics of the fn
’s play this role in our Padé examples. Due to S
, the‘‘Sturmians’’ may also be constructed. In the test evaluations of the binding energies and/or couplings, the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable.