### Abstract

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 *b* *a* *s* *i* *s* *p* *o* *t* *e* *n* *t* *i* *a* *l* *s* (powers, log, Hulthén, sech^{2}) 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* _{ n l } =*F* _{ n l }(*v*) of the Hamiltonian *H*=−Δ+*v* *f*(*r*), where *v* is a coupling constant. The transform Λ( *f*)=*F* is carried out in two steps: *f*→*f*̄→*F*, where *f*̄(*s*) is called the *k* *i* *n* *e* *t* *i* *c* *p* *o* *t* *e* *n* *t* *i* *a* *l* of *f* and is defined by *f*̄(*s*)=inf(ψ,*f*,ψ) subject to ψ∈D⊆*L* ^{2}(*R* ^{3}), where D is the domain of *H*, ∥ψ∥=1, and (ψ,−Δψ)=*s*. A table is presented of the basis kinetic potentials { *f*̄^{(i)}(*s*)}; the general trajectory bounds *F* _{*}(*v*) are then shown to be given by a Legendre transformation of the form **(** *s*, *f*̄_{*}(*s*)**)** →**(** *v*, *F* _{*}(*v*)**)**, where *f*̄_{*}(*s*) =∑^{ k } _{ i=1} *g* ^{(i)}× **(** *f*̄^{(i)}(*s*)**)** and *F* _{*}(*v*) =min_{ s>0}{*s*+*v* *f*̄_{*}(*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.’’

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