Spectral Theory of the Hermite Operator on L^p(Rⁿ)

Department of Mathematics and Statistics, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada

^⋆ Corresponding author. E-mail: duanxiao@yorku.ca. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

Abstract

We prove that the minimal operator and the maximal operator of the Hermite operator are the same on L^p(ℝⁿ) , 4 / 3 <p< 4. The domain and the spectrum of the minimal operator (=maximal operator) of the Hermite operator on L^p(ℝⁿ), 4/3 <p< 4, are computed. In addition, we can give an estimate for the L^p-norm of the solution to the initial value problem for the heat equation governed by the minimal (maximal) operator for 4/3<p<4.