Uncertainty Principles for Sturm–Liouville Operators

An uncertainty principle for the Sturm--Liouville operator $$ L=frac{d^2}{dt^2}+a(t)frac{d}{dt} $$ is established, as generalization of an inequality for Jacobi expansions proved in our previous paper, which implies the uncertainty principle for ultraspherical expansions by M. Rösler and M. Voit. The properties of the orthogonal set of eigenfunctions of the operator L and the so-called conjugate orthogonal set are unified by introducing the differential–difference operators, which are essential in our study. Asconsequences, an uncertainty principle for Laguerre, Hermite, and generalizedHermite expansions is obtained, respectively.