Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval

Kamil Kaleta
2012 Studia Mathematica  
We prove the uniform lower bound for the difference λ_2 - λ_1 between first two eigenvalues of the fractional Schrödinger operator, which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving open interval (a,b) ∈ with symmetric differentiable single-well potential V in the interval (a,b), α∈ (1,2). "Uniform" means that the positive constant appearing in our estimate λ_2 - λ_1 ≥ C_α (b-a)^-α is independent of the potential V. In general case of α∈ (0,2),
more » ... e also find uniform lower bound for the difference λ_* - λ_1, where λ_* denotes the smallest eigenvalue related to the antisymmetric eigenfunction ϕ_*. We discuss some properties of the corresponding ground state eigenfunction ϕ_1. In particular, we show that it is symmetric and unimodal in the interval (a,b). One of our key argument used in proving the spectral gap lower bound is some integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey-Lemma.
doi:10.4064/sm209-3-5 fatcat:fdtogsrosbfc3dwyhyvwtqea7e