Symmetry properties of quasiclassical energy levels

The quasiclassical ground-state energies exhibit typical analytic structures, as well as covariance properties, with respect to certain symmetry transformations. Such symmetries enable us to define corresponding classes of equivalent Hamiltonians. The scaling properties of the underlying phase-space quantum have also been established. Here we shall consider spherically symmetrical Hamiltonians like $p^2$/$m_0$+$C_{n1\mathrm{}}$ ${/}_1^n$+$C_{n2\mathrm{}}$ ${/}_2^n$, where $n_1$≠2, & (i=1,2) denote the couplings, whereas $n_i$ are the power exponents. Generalizations towards exact or approximate energy levels, depending on the values of $n_1$ and $n_2$, have also been performed. For $n_1$=2, this procedure leads us to reobtain the exact energy levels for $n_2$=1 and $n_2$=-2, and to propose closed estimates for the other $n_2$ values. The quasiclassical equivalence between the linear plus Coulomb potential and the quartic anharmonic oscillator has also been established.