Lie Theory and Separation of Variables for the Equation \[ iU_t + Δ _2 U - ( {{α }{{x_1^2 }} + {β }{{x_2^2 }}} )U = 0\]

This work constitutes a detailed study of the symmetries of the time-dependent Schrödinger equation in two spatial dimensions with an added inverse square potential of the form ((a / x₁² ) + (β / x₂² )). An intimate connection between these symmetries and the coordinate systems in which the equation separates is established. It is shown that there is a 1–1 correspondence between orbits of commuting pairs of symmetry operators—one taken from the Lie algebra of the symmetry group G and the other a second order symmetry operator—and G—inequivalent separable coordinate systems for the equation. The spectral analysis for all the basis functions corresponding to the different separable coordinate systems is computed. Then, making use of the symmetry group G, many addition and expansion theorems for the basis functions are derived. In this way, we find many special function identities involving Jacobi and Gegenbauer polynomials, Laguerre and Hermite polynomials, Whittaker functions, Bessel functions, parabolic cylinder functions, Airy functions, anharmonic oscillator functions, generalized spheroidal wave functions, generalized Ince functions and others. Many of these relations appear to be new.