The gap between the first two eigenvalues of a one-dimensional Schrödinger operator with symmetric potential

Abstract:

We prove the inequality ${\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]$ for the difference of the first two eigenvalues of one-dimensional Schrödinger operators $- \frac {{{d^2}}}{{d{x^2}}} + {V_i}(x),i = 0,1$, where ${V_1}$ and ${V_0}$ are symmetric potentials on $(a,b)$ and on $(a,(a + b)/2)$, and ${V_0} - {V_1}$ is decreasing on $(a,(3a + b)/4)$.