Energy levels for a double-well potential in three-dimensional system using Hill determinant approach (Q1919414)

Energy levels of the Schrödinger equation having a double-well potential \[ \begin{multlined} V(x, y, z; \lambda, Z^2)=\\ -\textstyle{{1\over 2}} Z^2[x^2+ y^2+ z^2]+ \textstyle{{1\over 2}} \lambda[a_{xx} x^4+ a_{yy} y^4+ a_{zz} z^4+ 2a_{xy} x^2 y^2+ 2a_{xz} x^2 z^2+ 2a_{yz} y^2 z^2]\end{multlined} \] in a three-dimensional system are calculated for some eigenstates \((n_x n_y n_z)\) and a wide range of the perturbation parameters \(Z^2\) and \(\lambda\). The Hill determinant approach is used. A recurrence relation for the double-well potential using the Hill determinant is derived. Plots are shown representing the energy levels of the states (000), (100), (110), and (111), at \(Z^2\) values \(0-100\), \(\lambda= 1\). Eigenvalues over a range of \(0\leq Z^2\leq 10^3\) and \(10^{- 1}\leq \lambda\leq 10^5\) for various eigenstates \((n_x n_y n_z)\) calculated by the Hill determinant approach are also tabulated.