Moment problem quantization within a generalized scalet-Wigner (auto-scaling) transform representation

Abstract

For one-dimensional Schrodinger quantum systems, the correlation expression S(x, τ, a) ≡ Ψ*(x − τ/2/a)Ψ(x + τ/2) satisfies a fourth-order linear differential equation with regard to x. This generalizes the result previously derived by Handy (2001 J. Phys. A: Math. Gen. 34 L271), and Handy and Wang (2001 J. Phys. A: Math. Gen. 34 8297), with regard to S(x, 0, 1). We are then able to incorporate this within a generalized Wigner transform representation, through the use of scalets (Handy C R and Brooks H A 2001 J. Phys. A: Math. Gen. 34 3577). Energy quantization is achieved through a moment problem positivity analysis (focusing on the moments of the probability density, Ψ2) at a = 1, τ = 0. The wavefunction, Ψ, is then generated through a multiscale analysis proceeding from the unity scale, extended into the zero scale limit. The power moments, μp = ∫ dx xp Ψ(x), can be generated through a similar procedure. We present the general formalism and apply it to the (ix)3 potential.