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

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator Aμ in L2(R, |t |2μ exp(−t2)), generated from the Dunkl second-order Hermite differential equation μ[y](t ) := −T 2 μ(y)(t )+ 2tTμ(y)(t )− 2μ y(t)− y(−t) + ky(t ) = λy (t ∈ R), that has the generalized Hermite polynomials {H μ m}∞m=0 as eigenfunctions and where Tμ 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 μ n and associated inner product (. , .)μ, n, which is generated from the nth integral power n μ[.] of μ[.]. Moreover, for each n ∈ N, we determine the corresponding unique left-definite self-adjoint operator Aμ,n in W μ n and characterize its domain in terms of another left-definite space. As a consequence, we explicitly determine the domain of each integral power of Aμ and in particular, we obtain a new characterization of the domain of the Dunkl right-definite operator Aμ.  2004 Elsevier Inc. All rights reserved.