高阶非谐振子的谱性质
Spectral properties of higher order anharmonic oscillators
摘要：
讨论了自伴算子\[ -d^2/dt^2 + (t^{k+1}/(k+1)-α)^2 \]在$L^2(\mathbb{R})$中的谱性质，其中自伴算子是奇数$k$ .我们证明了该算子的基态能量在$α$上的最小值是在一个唯一的点上达到的，当$k$趋于无穷大时，该点趋于零。此外，我们证明了最小值是非退化的。这些问题在磁场作用下薛定谔算子的谱分析中自然出现。这扩展或澄清了Pan-Kwek，Helffer-Morame，Aramaki，Helffer-Kordyukov和Helffer以前的结果。
英文摘要：
We discuss spectral properties of the self-adjoint operator \[ -d^2/dt^2 + (t^{k+1}/(k+1)-α)^2 \] in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over $α$ of the ground state energy of this operator is attained at a unique point which tends to zero as $k$ tends to infinity. Moreover, we show that the minimum is non-degenerate. These questions arise naturally in the spectral analysis of Schrödinger operators with magnetic field. This extends or clarifies previous results by Pan-Kwek, Helffer-Morame, Aramaki, Helffer-Kordyukov and Helffer.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
