= Sine trial bound for a linear potential on an interval
{title2=$\lambda_1(-\partial_x^2+\epsilon x)\leq\pi^2/L^2+\epsilon L/2$}
On $(0,L)$ with zero <Dirichlet boundary conditions>, the normalized trial $\sqrt{2/L}\sin(\pi x/L)$ has kinetic <Rayleigh quotient> $\pi^2/L^2$ and mean position $L/2$ by reflection symmetry. The <Rayleigh-Ritz variational principle> therefore gives the displayed upper bound for a real linear potential. For $\epsilon\geq0$ the lower bound $\lambda_1\geq\pi^2/L^2$ follows from the <Poincaré inequality>. For $\epsilon\ne0$ the trial is not an exact eigenfunction, so the upper bound is strict.
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