Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering

Abstract

Dedicated with great pleasure to Israel Samoilovich Kac on the occasion of his 85th birthday.Abstract. We find the high energy asymptotics for the singular Weyl-Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrö-dinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.

Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions

“…More precisely, let H be a singular Schrödinger operator on L 2 (a, b) as in [11] or [12] As a consequence we obtain that (4.12) holds at least for Im(t) < 0. To take the limit Im(t) → 0 we need the following result which follows from [ Then F (ε) = 1 4πi(t + iε) R e − p 2 4(t+iε) dα(p).…”

Dispersion estimates for spherical Schrödinger equations with critical angular momentum

“…is called the Weyl m-function (we refer to [16,18] for further details). Note that both f (k) and g(k) are analytic in the upper half plane and f (k) has simple zeros…”

“…Thus, by [18, Theorem 2.1] (see also Eq. (5.15) in [18] or [13]), on the real line we have |f (k)| = |k|(1 + o(1)), k → ∞.…”

On the Weyl solution of the 1-dim Schrödinger operator with inverse fourth power potential

“…[8,9,14,15,19]. Finally, we want to mention that generalised Titchmarsh-Weyl theory has also been extended to Sturm-Liouville operators with distributional and measure coefficients as well as to the one-dimensional Dirac operator and Jacobi operators, [2,[5][6][7]].…”

“…Remark The class of strongly singular potentials (to which 1 x 4 belongs) and their generalized Titchmarsh-Weyl coefficients have recently been studied in [10,11,13], see also [15] where more details on the earlier history can be found.…”