In this paper we study the perturbation \(L=H+V\), where \(H=-\frac{{{d}^{2m}}}{d{{x}^{2m}}}+{{x}^{2m}}\) on \(\mathbb {R}\), \(m\in {{\mathbb {N}}^{*}}\) and V is a decreasing scalar potential. Let \({{\lambda }_{k}}\) be the \(k^{th}\) eigenvalue of H. We suppose that the eigenvalues of L around \({{\lambda }_{k}}\) can be written in the form \({{\lambda }_{k}}+{{\mu }_{k}}\). The main result of the paper is an asymptotic formula for fluctuation \(\{ {{\mu }_{k}} \}\) which is given by a transformation of V. In the case \(m=1\) we recover a result on the harmonic oscillator.

中文翻译：

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标量电势下降引起的谐波振荡器

在本文中，我们研究摄动\（L = H + V \），其中\（H =-\ frac {{{d} ^ {2m}}} {d {{x} ^ {2m}}} + { {X} ^ {2米}} \）上\（\ mathbb {R} \），在\（米\ {{\ mathbb {N}} ^ {*}} \）和V是一个递减标量势。令\（{{\ lambda} _ {k}} \）为H的\（k ^ {th} \）特征值。我们假设\（{{\ lambda} _ {k}} \）周围的L的特征值可以用\（{{\ lambda} _ {k}} ++ } \）。本文的主要结果是波动的渐近公式\（\ {{{{mu} _ {k}} \} \）由V的变换给出。在\（m = 1 \）的情况下，我们在谐波振荡器上恢复结果。