H. Taşeli, İ. M. Erhan and Ö. Uğur, An Eigenfunction Expansion for the Schrödinger Equation with Arbitrary Non-Central Potentials, Journal of Mathematical Chemistry, 32(4), pp. 323-338, (November 2002).
An eigenfunction expansion for the Schrödinger equation for a particle moving in an arbitrary non-central potential in the cylindrical polar coordinates is introduced, which reduces the partial differential equation to a system of coupled differential equations in the radial variable \( r \). It is proved that such an orthogonal expansion of the wavefunction into the complete set of Chebyshev polynomials is uniformly convergent on any domain of \( (r, \theta) \). As a benchmark application, the bound states calculations of the quartic oscillator show that both analytical and numerical implementations of the present method are quite satisfactory.
Keywords: two-dimensional Schrödinger equation, eigenfunction expansion, eigenvalue problems