## Singularities of the Green function of the nonstationary Schrödinger equation.(English. Russian original)Zbl 0920.35008

Funct. Anal. Appl. 32, No. 2, 132-134 (1998); translation from Funkts. Anal. Prilozh. 32, No. 2, 80-83 (1998).
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 $$d>1$$.

### MSC:

 35A20 Analyticity in context of PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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### References:

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