Analytic treatment of the αx2+βx4 oscillator

Abstract

We carry out an investigation of the Schrödinger equation with the αx2+βx4 interaction for 0<Re( x)<∞, and β=‖β‖ exp(i(π+λ)),−π<λ<π. In the sectors 0≤arg( x)<π/3−λ/6, 0≤λ<π, and −λ/6<arg( x)<π/3−λ/6, −π<λ<0, a subdominant solution is constructed essentially in terms of the Laplace transform of a function f (s), 0≤s<∞, which is expressible as a converging power series in s. The solution, y( x), thus obtained, is compared and contrasted with the one valid in the usual case where Re( x)≤0 is included and it is argued that y( x) may not be accessible to perturbative approaches.