The left-definite spectral theory for the classical Hermite differential equation

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A in L2((−∞,∞);exp(−t2)), generated from the classic second-order Hermite differential equationℓH[y](t)=−y″+2ty′+ky=λy (t∈(−∞,∞)),that has the Hermite polynomials {Hm(t)}m=0∞ as eigenfunctions. More specifically, for each n∈N, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn and associated inner product (·,·)n, which is generated from the nth integral power ℓHn[·] of ℓH[·]. Moreover, for each n∈N, we determine the corresponding unique left-definite self-adjoint operator An in Wn and characterize its domain in terms of another left-definite space. As a consequence of this, we explicitly determine the domain of each integral power of A and, in particular, we obtain a new characterization of the domain of the classical right-definite operator A.