WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator

An investigation is made of approximate solutions of three-term recursion relations with slowly varying coefficients that have a form similar to the WKB solutions of second-order differential equations. Potential curves for a recursion relation are introduced, and the matching conditions at simple and singular turning points are analyzed. Quantization rules (of the Bohr–Sommerfeld type) are formulated, and this makes it possible to find the eigenvalues and eigenvectors of tridiagonal matrices. The method is applied to the analysis of the equation that describes the state of an anharmonic oscillator in a resonant periodic field.