Finite diagonal lift and its relative frequency window

Fix a spin label $J\ge 1$ and the spin blocks $V_J$ and $V_{J-1}$. Let $H_0$ and $D$ be diagonal in their combined orthonormal spin basis, with real eigenvalues $E_{Kn}$ and $d_{Kn}=\mathrm{\Delta }Kn$, respectively, for $K\in \{J,J-1\}$ and $n=-K,-K+1,\dots ,K$, where $\mathrm{\Delta }$ is real. Use units with $\mathrm{\hslash }=1$, so directed frequencies are source energies minus destination energies.

For every formal triple with $m=-J,-J+1,\dots ,J$ and $\delta \in \{-1,0,1\}$, define $s_{Jm\delta }=\mathrm{\Delta }[m-(J-1)\delta ]$. Then $\mathrm{\mid }{s}_{Jm\delta }\mathrm{\mid }\le (2J-1)\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$. The pair $(J,m)\to (J-1,m+\delta )$ designates existing spin states only when $m+\delta \in \{-(J-1),-(J-1)+1,\dots ,J-1\}$. For such a pair, $s_{Jm\delta }$ is the change of its directed frequency on replacing $H_0$ by $H_0+D$. This is an energy-difference statement and does not assert that a transition matrix element is nonzero; for labels without a destination state, $s_{Jm\delta }$ remains only a formal algebraic expression.

If the transitions under consideration have unperturbed directed frequency $E_{Jm}-E_{J-1,m+\delta }={\mathrm{\Omega }}_J={\omega }_0{h}_J$, where ${\omega }_0,h_J>0$, then $\mathrm{\mid }{s}_{Jm\delta }\mathrm{/}{\mathrm{\Omega }}_J\mathrm{\mid }\le r_J=(2J-1)\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}({\omega }_0{h}_J)$. For $r_J<1$, all their shifted frequencies ${\mathrm{\Omega }}_J+s_{Jm\delta }$ remain positive. The relative bound and positivity of the formal numbers ${\mathrm{\Omega }}_J+s_{Jm\delta }$ also hold for every listed formal triple.