We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ${(-\Delta )}^{\alpha /2}+V$, α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_{\alpha }$ appearing in our estimate $\lambda ₂-\lambda ₁\ge C_{\alpha }{(b-a)}^{-\alpha }$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for...

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$. Restrict it to some large cube $\Lambda$. Consider now $I_{\Lambda }$, a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $\Lambda$ grows to infinity. We prove that, with probability one in the large volume...