Chao-Zhong Chen et al. $[{ Proc}.$ ${ Amer. Math. Soc},2013],$ proved the upper estimate $\frac{ \lambda _{ n}}{ \lambda _{ m}}\leq \frac{ % n^{ p}}{ m^{ p}}$ $ (n>m\geq 1) $ for Dirichlet Shr\"{ o}dinger operators with nonnegative and single-well potentials. In this paper we discuss the case of nonpositive potentials $q(x)$ continuous on the interval $[ 0,1] $. We prove that if $q(x)\leq 0$ and single-barrier then $\frac{ \lambda _{ n}}{ \lambda _{ m}}\geq \frac{ n^{ p}% }{ m^{ p}}$ for $\lambda _{ n}>\lambda _{ m}\geq -2q^{ \ast },$ where $q^{ \ast}=\inf\{ q(0), q(1)\}$. Furthermore, we show that there exists $\ell_{ 0}\in ( 0,1] $ such that for all $\ell\in(0,\ell_{ 0}],$ the associated eigenvalues $(\lambda _{ n}(\ell)) _{ n\geq 1}$ (of the problem defined on $[0,\ell]$) satisfy $ \lambda _{ 1}( \ell)>0$ and $\frac{ \lambda _{ n}( \ell)}{ \lambda _{ m}( \ell) }\geq \frac{ n^{ p}}{ m^{ p}}$ $n>m\geq 1$. The value $\ell _{ 0}$ satisfies the following estimate $0<\ell_{ 0}\leq \sqrt[p]{ \frac{ -p}{ 3q^{ *}}}$.
Categories: math.SP