A geometrical theory of energy trajectories in quantum mechanics

Suppose f(r) is an attractive central potential of the form f(r)=∑^k_i=1 g⁽ⁱ⁾( f⁽ⁱ⁾(r)), where {f⁽ⁱ⁾} is a set of basis potentials (powers, log, Hulthén, sech²) and {g⁽ⁱ⁾} is a set of smooth increasing transformations which, for a given f, are either all convex or all concave. Formulas are derived for bounds on the energy trajectories E_nl =F_nl(v) of the Hamiltonian H=−Δ+vf(r), where v is a coupling constant. The transform Λ( f)=F is carried out in two steps: f→f̄→F, where f̄(s) is called the kinetic potential of f and is defined by f̄(s)=inf(ψ,f,ψ) subject to ψ∈𝒟⊆L²(R³), where 𝒟 is the domain of H, ∥ψ∥=1, and (ψ,−Δψ)=s. A table is presented of the basis kinetic potentials { f̄⁽ⁱ⁾(s)}; the general trajectory bounds F_*(v) are then shown to be given by a Legendre transformation of the form (s, f̄_*(s)) →(v, F_*(v)), where f̄_*(s) =∑^k_i=1g⁽ⁱ⁾× ( f̄⁽ⁱ⁾(s)) and F_*(v) =min_s>0{s+v f̄_*(s)}. With the aid of this potential construction set (a kind of Schrödinger Lego), ground‐state trajectory bounds are derived for a variety of translation‐invariant N‐boson and N‐fermion problems together with some excited‐state trajectory bounds in the special case N=2. This article combines into a single simplified and more general theory the earlier ‘‘potential envelope method’’ and the ‘‘method for linear combinations of elementary potentials.’’