Semiclassical energy formulas for power-law and log potentials in quantum mechanics

We study a single particle which obeys non-relativistic quantum mechanics in R and has Hamiltonian H = −∆ + V (r), where V (r) = sgn(q)r. If N ≥ 2, then q > −2, and if N = 1, then q > −1. The discrete eigenvalues Enl may be represented exactly by the semiclassical expression Enl(q) = minr>0{Pnl(q) /r + V (r)}. The case q = 0 corresponds to V (r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the Pnl(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q) = Z(q)P (q) is monotone increasing, even though the factor Z(q) = (1+q/N) is monotone decreasing. Thus P (q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.