Affine-parameter Sturm-Liouville eigenvalue bound (${\lambda }_{\min }(ϵ)\le {\lambda }_{\min }(0)+ϵ⟨U_0,L_1{U}_0⟩$)

For a self-adjoint Sturm-Liouville operator $L_{ϵ}=L_0+ϵL_1$ with common boundary conditions and an $L^2$-normalized lowest eigenfunction $U_0$ of $L_0$, evaluating the Rayleigh quotient on $U_0$ gives the displayed upper bound. More generally the lowest eigenvalue is the infimum of affine functions of $ϵ$, so it is concave in that parameter. The trial state must satisfy the boundary requirements of the variational domain; changing the operator without changing the trial function's normalization can still be legitimate when the weight is fixed.