Solution of the Heisenberg equations of motion for a double-well potential. Matrix elements and the energy eigenvalues

The Heisenberg equations of motion are applied to find the energy levels of the symmetric double-well potential (- 1/2 x2 + λ/4 x4). Following Klein's method of approximating infinite matrices by finite matrices, one can reduce the eigenvalue problem to that of finding the roots of a set of nonlinear algebraic equations. In general, there are sets of roots satisfying this truncated group of equations, but among these, the solution where the eigenvalues are well-ordered gives accurate results particularly for low-lying states