Singularities of the Green function of the nonstationary Schrödinger equation
The Green function \(\psi= \psi(x,y,t)\) of the nonstationary Schrödinger equation is determined by the relations \[ ih\psi_t= -\textstyle{{1\over 2}} h^2\Delta\psi+ v(x)\psi,\quad \psi(x,y,0)= \delta(x- y),\;t\geq 0,\;y\in\mathbb{R}^d. \] In what follows, we assume that \(v\) is a smooth real function. For a chosen \(t>0\), the function \(\psi\) can have singularities with respect to \(x\). These singularities depend on the rate of growth of \(v(x)\) as \(x\to\infty\). If \(v\) satisfies the condition \[ | v(x)|\leq c| x|^{2\gamma},\quad \gamma<1, \] for sufficiently large \(x\), then \(\psi\) is smooth. This assertion, as well as several subsequent assertions, is given here in a simplified form. The main objective of the present paper is to describe the formula mechanism responsible for the form of the singularities of \(\psi\) for \(\gamma>1\) in the case \(d=1\). Under certain restrictions, we shall be able to say something about the character of the singularities of \(\psi\) for \(d>1\).
scientific article; zbMATH DE number 1267936