Dispersion estimates for spherical Schrödinger equations with critical angular momentum

Abstract

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators, where the angular momentum takes the critical value l = − 1 2 . We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.2010 Mathematics Subject Classification. Primary 35Q41, 34L25; Secondary 81U30, 81Q15.

A transmutation operator method for solving the inverse quantum scattering problem *

“…is called the Jost solution. The uniqueness and the existence of both regular and Jost solutions is a well known fact (see, e.g., [5], and for non-integer values of , [13,19] and references therein).…”

Transformation operators for spherical Schrödinger operators

“…The term ∂B(x,y) ∂y y=0 φ l (z, 0) disappears, since φ l (z, y) = O(y l+1 ) by the properties of φ l mentioned e.g. in [10,16,Section 2], and ∂B(x,y) ∂y can be assumed to be bounded(cf. Lemma 2.12).…”

“…It's also worthwhile mentioning, that one field of recent research is concerned about proving dispersive estimates for the related Schrödinger equations, c.f. [9], [10], [16] and [17]. In many of these contributions the existence and precise estimates for transformation operators for H are crucial.…”

“…Concerning the asymptotic behavior of these solutions φ l , we refer e.g. to [10,16,Section 2]. We want to express this transformation operator as an integral operator and prove an estimate for it.…”

“…establishing the following theorem, where f (k, x), f l (k, x) denote the Jost solutions of the corresponding equations (1.2), (1.3) respectively, which satisfy f (k, x) ∼ e ikx as x → ∞(for details, again cf. [10,16,Section 2]):…”