The one- and two-dimensional quartic oscillators

Accurate eigenvalues and eigenvectors of the one- and two- dimensional quartic oscillators have been determined using the linear variation method in which the Hamiltonian matrix was set up in the representation of the corresponding harmonic oscillator. The Hamiltonian matrix was factorized and the submatrices were diagonalized using two different methods: (i) Matrices of order 100 were diagonalized giving both eigenvalues and eigenvectors. (ii) Matrices of order 800 were diagonalized giving eigenvalues only. The former of these procedures yields eigenvalues accurate to nine figures. Mixed harmonic-quartic potentials have been investigated for the two-dimensional case.