A Representation Formula Related to Schrödinger Operators

Let H = −d/dx + V be a Schrödinger operator on the real line, where V ∈ L ∩ L. We define the perturbed Fourier transform F for H and show that F is an isometry from the absolute continuous subspace onto L(R). This property allows us to construct a kernel formula for the spectral operator φ(H). Schrödinger operator is a central subject in the mathematical study of quantum mechanics. Consider the Schrödinger operator H = −△+ V on R, where △ = d/dx and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of −△. A natural conjecture is that an L function admits a similar expansion in terms of “eigenfunctions” of H , a perturbation of the Laplacian (see [7]. Ch.XI and the notes), under certain condition on V . The three dimension analogue was proven true by T.Ikebe [6], a member of Kato’s school, in 1960. Later his result was extended by Thor to the higher dimension case [10]. In one dimension, recent related results can be found in e.g., Guerin-Holschneider [5], Christ-Kiselev [4] and Benedetto-Zheng [3]. Throughout this paper we assume V : R → R is in L ∩L. We shall prove a one-dimensional version of Ikebe’s theorem for L functions (Theorem 1). Theorem 2 presents an integral formula for the kernel of the spectral operator φ(H) for a continuous function φ with compact support. In a sequel to this paper we shall use this explicit formula to study function spaces associated with H (see [3]). The generalized eigenfunctions e(x, ξ), ξ ∈ R of H satisfy (1) (−d /dx + V (x))e(x, ξ) = ξe(x, ξ) in the sense of distributions. Date: February 8, 2008. 2000 Mathematics Subject Classification. Primary: 42C15; Secondary: 35P25.