The left-definite spectral theory for the Dunkl–Hermite differential-difference equation

Abstract

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A μ in L 2( R,|t| 2μ exp(−t 2)) , generated from the Dunkl second-order Hermite differential equation ℓ μ[y](t):=−T μ 2(y)(t)+2tT μ(y)(t)−2μ y(t)−y(−t) +ky(t)=λy (t∈ R), that has the generalized Hermite polynomials { H m μ } m=0 ∞ as eigenfunctions and where T μ is a differential-difference operator called the Dunkl operator on R of index μ. More specifically, for each n∈ N , we explicitly determine the unique left-definite Hilbert space W n μ and associated inner product (. ,.) μ, n , which is generated from the nth integral power ℓ μ n [.] of ℓ μ [.]. Moreover, for each n∈ N , we determine the corresponding unique left-definite self-adjoint operator A μ, n in W n μ and characterize its domain in terms of another left-definite space. As a consequence, we explicitly determine the domain of each integral power of A μ and in particular, we obtain a new characterization of the domain of the Dunkl right-definite operator A μ .