Asymptotic of the limit phases in a problem of potential scattering (Q808313)

Under the assumption \[ (1)\quad q(x)\leq C(1+| x|)^{-b},\quad b>1,\quad q=\bar q, \] the solution of the Schrödinger equation \[ (2)\quad -\Delta u+q(x)u=\lambda u,\quad \lambda >0,\quad x\in {\mathbb{R}}^ d,\quad d\geq 2, \] with the asymptotic type standard wave \[ (3)\quad u(x)\sim \psi (\omega)r^{-(d-1)/2} \sin (\lambda^{1/2}r-4^{-1}(d- 3)\pi -\theta),\quad \psi \in L_ 2(S^{d-1}),\quad x=r\omega, \] as \(r=| x| \to \infty\) can be given for some (\(\theta\),\(\psi\) (\(\omega\))). Possible phases and functions (\(\theta\),\(\psi\) (\(\omega\))) can be characterized by the unitary diffusion matrix S in \(L_ 2(S^{d-1})\) from equation (2) with a compact operator S-I. (\(\theta\),\(\psi\)) suitable to the equation (2) can be given by \(SJ\psi (\omega)\equiv S\psi (- \omega)=\mu \psi (\omega)\) and \(\mu =\exp (-2i\theta)\). \(\theta\) is described as follow: \(\exp (\mp 2i\delta^{\pm}_ n)\), \(0<\delta^{\pm}_{n+2}\leq \delta^{\pm}_ n<\pi /4\) on the right semicircle; and \(-\exp (\mp 2i\pi^{\pm}_ n)\), \(0<\eta^{\pm}_{n+1}\leq \eta^{\pm}_ n<\pi /4\) on the left semicircle. The author gives the following result on the limit of \(\{\delta^{\pm}_ n\}\) and \(\{\eta^{\pm}_ n\}:\) Suppose that the symmetric part \(q_ s(x)=2^{-1}(q(x)+q(-x))\) satisfies the condition \(q_ s(x)=O(| x|^{-a})\) for some \(a>1\) and that the estimation (1) holds for \(b>2^{-1}(a+1).\) Then \(\delta^{\pm}_ n=O(n^{-\rho})\) and \(\eta^{\pm}_ n=O(n^{- \rho})\) hold, where \(\rho =(a-1)(d-1)^{-1}.\) Furthermore under the assumptions \[ q_ s(x)=| x|^{- a}g(x| x|^{-1})+o(| x|^{-a}),\quad | x| \to \infty,\quad g\in C^{\infty}(S^{d-1}), \] he obtains the value \(\lim_{n\to \infty}n^{\rho}\delta^{\pm}_ n=\lim_{n\to \infty}n^{\rho}\eta^{\pm}_ n\). Finally he shows the essential part of his proof as a theorem.