Abstract
In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(mu) in L-2(R, \t\(2mu) exp(-t(2))), generated from the Dunkl second-order Hermite differential equation
l(mu)[y](t) := -T-mu(2)(y)(t) + 2tT(mu)(y)(t) - 2mu(y(t) - y(t) - y(-t)) + ky(t) = lambday (t is an element of R),
that has the generalized Hermite polynomials {H-m(mu)}(m=0)(infinity) as eigenfunctions and where T-mu is a differential-difference operator called the Dunkl operator on R of index mu. More specifically, for each n is an element of N, we explicitly determine the unique left-definite Hilbert space W-n(mu) and associated inner product (.,.)mu,n, which is generated from the nth integral power l(mu)(n)[.] of l(mu)[.]. Moreover, for each n is an element of N, we determine the corresponding unique left-definite self-adjoint operator A(mu,n) in W-n(mu) 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(mu) and in particular, we obtain a new characterization of the domain of the Dunkl right-definite operator A(mu). (C) 2004 Elsevier Inc. All rights reserved.