Spectral Decomposition of the Weinstein Laplacian and Its Application to the Associated Heat Transform

Abstract

For the Weinstein Laplacian considered on the Hilbert space which makes it a self‐adjoint operator, the Von Neumann spectral decomposition is given. As applications, a new integral representation for the Weinstein heat kernel is given. Also, it is proved that the spectrum of the semigroup associated with the Weinstein Laplacian is reduced to its continuous spectrum which is given by the interval [0, 1]. Moreover, it is proved that each λ in [0, 1] is a generalized eigenvalue associated with a generalized eigenvector.