Schrödinger's equation and continued fractions

Various connections between Jacobi matrices, continued fractions, orthogonal polynomials, three‐term recurrence relationships, and the time‐independent Schrödinger equation are reviewed and applied to obtain bound states and/or resonances for (1) The inverted oscillator Hamiltonian (1/2) ( p 2 − r 2 ) (2) The Coulomb plus oscillator Hamiltonian p 2 − zr −1 + ( r + β) 2 /4α 2(3) A general Hamiltonian P 2 + V ( r ) in the lattice approximation with explicit examples for V ( r ) = − z / r or λ r It is pointed out that a scattering theory follows naturally from the properties of the subdominant boundary value solution to the associated recurrence relationship. As an example we calculate the lattice Coulomb s wave phase shift.