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

Abstract

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A in L 2((−∞,∞);exp(− t 2)), generated from the classic second-order Hermite differential equation ℓ H[y](t)=−y″+2ty′+ky=λy (t∈(−∞,∞)), that has the Hermite polynomials { H m ( t)} m=0 ∞ as eigenfunctions. More specifically, for each n∈ N , we explicitly determine the unique left-definite Hilbert–Sobolev space W n and associated inner product (·,·) n , which is generated from the nth integral power ℓ H n [·] of ℓ H [·]. Moreover, for each n∈ N , we determine the corresponding unique left-definite self-adjoint operator A n in W n 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.