Anharmonic oscillator and the analytic theory of continued fractions

We study anharmonic oscillators of the type $a{x}^2+b{x}^4+c{x}^6$ using the theory of continued fractions. Introducing a new set of coupling constants (depending on $a$, $b$, and $c$) in terms of which the associated difference equation simplifies, we write the Green's function of the theory in terms of an infinite continued fraction of the Stieltjes type, whose poles give the energy eigenvalues. We prove that this continued fraction converges where the corresponding perturbation series in the dominant coupling diverges. We obtain the analytic structure of the Green's function in the complex plane of this coupling constant. A scale transformation allows us to study the analyticity of the Green's function for $a{x}^2+c{x}^6$ oscillators in the energy plane.