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- W1992808889 abstract "We study a single particle which obeys non-relativistic quantum mechanics in ℜN and has Hamiltonian H = −Δ + V(r), where V(r) = sgn(q)rq. 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)2/r2 + 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)1/q is monotone decreasing. Thus, P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvalues at the bottom of each angular momentum subspace." @default.
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- W1992808889 date "2003-06-11" @default.
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- W1992808889 title "Semiclassical energy formulae for power law and log potentials in quantum mechanics" @default.
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