Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval (Q2891269)

The author proves a lower bound for the difference \(\lambda_2-\lambda_1\) of the first two eigenvalues of the fractional Schrödinger operator \((-\Delta)^{\alpha/2}+V\), \(\alpha\in(1,2)\), with potential \(V\) in a bounded interval \((a,b)\), exploiting a Feynman-Kac representation of the semigroup of a symmetric \(\alpha\)-stable process on \((a,b)\) killed when leaving the interval and a certain integral inequality which is known to be a consequence of the Garsia-Rodemich-Rumsey lemma. The potential \(V\) is supposed to be symmetric (\(V(x)=V(a+b-x)\)), single well (\(V\) is nonincreasing on \((a,(a+b)/2)\)) and satisfies a Kato class type integrability condition. The lower bound, of the form \(\lambda_2-\lambda_1\geq C_\alpha (b-a)^{-\alpha}\), is uniform because the constant \(C_\alpha\) does not depend on the potential \(V\).NEWLINENEWLINEIn the general case \(\alpha\in (0,2)\), a similar uniform bound for the difference \(\lambda_*-\lambda_1\), where \(\lambda_*\) denotes the smallest eigenvalue of an antisymmetric eigenfunction, is also found.