Schr\"odinger equation. The theorem concerning the ansatz representation of a~solution concentrated in a~neighborhood of a~minimum of the potential

T. F. Pankratova

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 14, Tome 140 (1984), pp. 137-150

Résumé

The one-dimensional Schrödinger equation $-\frac{\hbar^2}{2m}y''+v(x)=F(y)$ is considered on the segment $[-l,l]$. It is assumed that the potential $v(x)$ of this equation has one minimum $v(0)=v'(0)=0$, $v''(0)>0$, $v(x)>0$ for $x\ne0$; $v(x)\ge h>0$ outside some neighborhood of zero. It is proved that there exists a solution of the form $\frac1{\sqrt{\psi'(x)}}D_n(\frac{\psi (x)}{\sqrt\hbar})$ where $D_n$ is a parabolic cylinder function, and $\psi$ is a smooth function which is bounded on $[-l,l]$ together with derivatives through third order by a constant not depending on $\hbar$. The function $\psi$ and the real number $E$ admit a known asymptotic expansion as $\hbar\to0$.