Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator
Qi Huang Yu, Jia Qing Zhong · Transactions of the American Mathematical Society · 1986
In this paper the authors prove the following theorem: Let Ω \Omega be a smooth strictly convex bounded domain in R n {R^n} and V : Ω → R V:\Omega \to R a nonnegative convex function. Suppose λ 1 {\lambda _1} and λ 2 {\lambda _2} are the first and second nonzero eigenvalues of the equation \[ − Δ f + V f = λ f , f | ∂ Ω ≡ 0. - \Delta f + Vf = \lambda f,\qquad f{|_{\partial \Omega }} \equiv 0. \] Then λ 2 − λ 1 ⩾ π 2 / d 2 {\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2} , where d d