The energy levels of the Schrödinger equation for the potentialx²+y²+λ[a_xxx⁴+2a_xyx²y²+a_yyy⁴] are calculated using Hill determinant approach for several eigenstates and over a wide range of values of perturbation parameters. The obtained numerical results agree with those previously reported by other methods.

Eigenenergies are calculated for the potentialsV₁(r)=−(a/r)[1+(1+br)e^−2br] andV₂(r)=−(v/r)[1 −λr(1−Z⁻¹)(1+λr)⁻¹], using renormalized series technique. Accurate results produced here for various eigenstates agree with those available in the literature.

The Schrödinger equation with the potentialV(r)=D[exp(−2β(r−r_e))−2exp(−β(r−r_e))] is treated in the framework of the hypervirial-renormalization parameter scheme. The energy eigenvalues of various eigenstates for different molecules are calculated.

The energy levels of a two-dimensional system are calculated for the rational potential,V(x, y; λ, g)=x²+y²+λ[x²/(1+gx²)+y²/(1+gy²)+a_xxx⁴+a_xyx²y²+a_yyy⁴] using the inner product technique over a wide range of values of the perturbation parameters (g, λ) and for various eigenstates.