Special analytical solutions of the Schrodinger equation for two and three electrons in a magnetic field and ad hoc generalizations to N particles

We found that the two-dimensional Schrodinger equation for three electrons in a homogeneous magnetic field (perpendicular to the plane) and a parabolic scalar confinement potential (frequency omega(0)) has analytical solutions in the limit where the expectation value of the centre-of-mass vector R is small compared with the average distance between the electrons. These analytical solutions exist only for certain discrete values of the effective frequency <(omega)over tilde> = root[omega(0)(2) + (omega(c)/2)(2)]. Furthermore, for finite external fields, the total angular momenta must be M-L = 3m with m = integer, and spins have to be parallel. The analytically solvable states are always cusp states, and take the components of higher Landau levels into account. These special analytical solutions for three particles and the exact solutions for two particles (Taut M 1994 J. Phys. A: Math. Gen. 27 1045 and Taut M 1994 J. Phys. A: Math. Gen. 27 4723 (erratum)) can be written in a unified form. The first set of solutions reads Phi = Pi(i(n,m) Sigma(l) r(l)(2)) where the p(n,m)(x) are certain finite polynomials and <(omega)over tilde>(n,m) is the spectrum of the fields. The pair angular momentum m has to be an odd integer and the integer n defines the number of terms in the polynomials. For infinite solvable fields <(omega)over tilde>(1) there is a second set of the form Phi = A(a) Pi(i(1) Sigma(l) r(l)(2)) where A(a) is the antisymmetrizer and the pair angular momenta mik can all be different integers. In both cases the first factor is a short-hand form with the convention r(m) = r(\m\)e(im alpha). These formulae, when ad hoc generalized to N coordinates, can be discussed as an ansatz for the wave function of the N-particle system. This ansatz fulfils the following requirements: it is exact for two particles and for three particles in the limit of small R and for the solvable external fields, and it is an eigenfunction of the total orbital angular momentum. The Laughlin functions are special cases of this ansatz for infinite solvable fields and equal pair angular momenta.