Schrödinger's equation with linear combinations of elementary potentials

Suppose $\epsilon =F\left(v\right)\mathrm{}$ is the energy of the lowest bound eigenstate of the Hamiltonian $\hat{h}=-\Delta +vf\left(x\right)\mathrm{}$, where $f\left(x\right)\mathrm{}$ is an attractive central potential. The $\Lambda$ transform is defined by $F=\Lambda \mathrm{}\left\{f\right\}\mathrm{}$ and the curve ($v$,$F\left(v\right)\mathrm{}$) is called the "energy trajectory" of the problem. In this article it is shown that the energy trajectory of the linear combination $f=\Sigma {i=1\mathrm{}}^n{\alpha }_i{f}_i$, ${\alpha }_i\ge 0$, is bounded by curves of the form $v={\left\{\Sigma {i=1\mathrm{}}^n{\alpha }_i{\left[{\theta }_i\mathrm{}\right(s\left)\right]}^{-1\mathrm{}}\right\}}^{-1\mathrm{}}$, $\epsilon =v\Sigma {i=1\mathrm{}}^n{\alpha }_i{F}_i\left({\theta }_i\mathrm{}\right(s\left)\right){\left[{\theta }_i\mathrm{}\right(s\left)\right]}^{-1\mathrm{}}$, where the curve parameter $s>\mathrm{}0\mathrm{}$, and each function ${\theta }_i\mathrm{}\left(s\right)\mathrm{}$ is defined by an equation $F_i\left({\theta }_i\right)-{\theta }_i{F}_i^{\prime }\left({\theta }_i\right)=s$, which is solvable when $f_i$ is "elementary." For the lower bound, $F_i=\Lambda \left\{f_i\right\}$, for the upper bound, $F_i\mathrm{}\left(v\right)=F_i^U\mathrm{}\left(v\right)\mathrm{}$ is the upper trajectory obtained by applying a trial function to the one-component problem $-\Delta +v{f}_i\mathrm{}\left(x\right)\mathrm{}$ and minimizing $〈\hat{h}〉$ with respect to a scale variable. Detailed recipes are given for the trajectory bounds corresponding to potentials which are arbitrary linear combinations of powers, logarithm, Hulthén, and ${\mathrm{sech}}^2$ in one or three dimensions. For the anharmonic oscillator