Kinetic potentials in quantum mechanics

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Abstract
Suppose that the Hamiltonian H=−Δ+vf(r) represents the energy of a particle which moves in an attractive central potential and obeys nonrelativistic quantum mechanics. The discrete eigenvalues Enl=Fnl(v) of H may be expressed as a Legendre transformation Fnl(v)=mins≳0(s+vf̄nl(s)), n=1,2,3,..., l=0,1,2,..., where the ‘‘kinetic potentials’’ f̄nl(s) associated with f(r) are defined by f̄nl(s) =infDnl supψ∈Dnl, ∥ψ∥=1 ∫ ψ(r) f ([ψ,−Δψ)/s]1/2r)ψ(r)d3r, and Dnl is an n-dimensional subspace of L2(R3) labeled by Ylm(θ,φ), m=0, and contained in the domain 𝒟(H) of H. If the potential has the form f(r)=∑Ni=1 g(i)( f(i)(r)) then in many interesting cases it turns out that the corresponding kinetic potentials can be closely approximated by ∑Ni=1 g(i)( f̄nl(i)(s)). This nice behavior of the kinetic potentials leads to a constructive global approximation theory for Schrödinger eigenvalues. As an illustration, detailed recipes are provided for arbitrary linear combinations of power-law potentials and the log potential. For the linear plus Coulomb potential and the quartic anharmonic oscillator the approximate eigenvalues are compared to accurate values found by numerical integration.