Small-energy asymptotics for the Schrödinger equation on the line (Q2747514)

The authors consider the one-dimensional Schrödinger equation NEWLINE\[NEWLINE-\psi''(k,x)+ V(x)\psi(k,x)= k^2\psi(k,x)NEWLINE\]NEWLINE on the line, where \(V\) is assumed to have a first moment. Solutions \(f\) of this equation, which satisfy NEWLINE\[NEWLINEe^{ikx} f_l(k,x)= 1+o(1),\quad e^{-ikx}f_l' (k,x)=ik+o(1) \quad\text{for }x\to \inftyNEWLINE\]NEWLINE and NEWLINE\[NEWLINEe^{ikx} fr(k,x)=1+o(1), \quad e^{ikx}fr'(k,x) =-ik+o(1)\quad \text{for }x \to-\inftyNEWLINE\]NEWLINE are called the left and right Jost solutions. They are important because scattering, reflection and transmission of waves can be expressed in terms of these functions. Their main result, for which two independent proofs are given, states for the small energy asymptotics: NEWLINENEWLINENEWLINETheorem. Assume \(V\) is real valued and has a first moment. For any \(x\in\mathbb{R}\) the Jost solutions \(f_\alpha\) ; \(\alpha=l,r\) satisfyNEWLINENEWLINENEWLINE1. If \(f_\alpha(0,x)\neq 0\) then \({f_\alpha'(k,x)\over f_\alpha (k,x)}={f_\alpha' (0,x)\over f_\alpha (0,x)}+\varepsilon (\alpha){ik \over f_l(0,x)^2} +o(k)\).NEWLINENEWLINENEWLINE2. If \(f_\alpha'(0,x)\neq 0\) then \({f_\alpha (k,x) \over f_\alpha' (k,x)}={f_\alpha (0,x)\over f_\alpha'(0,x)}-i\varepsilon (\alpha) {k\over f_\alpha'(0,x)^2} +o(k)\) with \(k\to 0\) in the upper half plane and \(\varepsilon (l)=+1\), \(\varepsilon (r)=-1\). If \(V\) has a second moment these results can be improved. The proofs are mainly based on estimates of the corresponding integral equation.