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Schrödinger's equation and continued fractions
Author(s) -
Masson David R.
Publication year - 1987
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560320769
Subject(s) - hamiltonian (control theory) , coulomb , schrödinger equation , boundary value problem , lattice (music) , physics , quantum mechanics , mathematical physics , wave function , orthogonal polynomials , scattering , mathematics , quantum electrodynamics , mathematical analysis , mathematical optimization , acoustics , electron
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.