Explicit formula for Schroedinger wave operators on the half-line for potentials up to optimal decay

2019/03/11 by Hideki Inoue, Inoue, Hideki
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1903.04242

Abstract

We give an explicit formula for the wave operators for Schroedinger operators on the half-line with a potential decaying strictly faster than the polynomial of degree minus two. The formula consists of the main term given by the scattering operator and a function of the generator of the dilation group, and a Hilbert-Schmidt remainder term. Our method is based on the elementary construction of the generalized Fourier transform in terms of the solutions of the Volterra integral equations. As a corollary, a topological interpretation of Levinson's theorem is established via an index theorem approach.

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