Abstract

A first-order regularized trace formula has been obtained for the Sturm–Liouville operator with a point of \( \delta ^{\prime }\)-interaction. For large values of the spectral parameter, asymptotic representations have been found for solutions of the Sturm–Liouville equation with discontinuity conditions. The asymptotics of the eigenvalues of the operator under study has been derived. It is shown that there appears an additional term in the regularized trace formula that takes into account the jump in the charge distribution function in the middle of the interval. Note that regularized traces are used to approximately calculate the first eigenvalues of the operator under consideration. These traces are also useful when solving inverse spectral analysis problems for differential equations.

中文翻译：

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具有 $$\delta ^{\prime }$$ -Interaction Point 的 Sturm-Liouville 算子的迹公式

摘要

已经获得了 Sturm-Liouville 算子的一阶正则化迹公式，其中点为\( \delta ^{\prime }\) -相互作用。对于大的谱参数值，已经找到了具有不连续性条件的 Sturm-Liouville 方程解的渐近表示。已经导出了所研究算子的特征值的渐近线。结果表明，在正则化迹公式中出现了一个附加项，它考虑了区间中间电荷分布函数的跳跃。请注意，正则化迹线用于近似计算所考虑的算子的第一个特征值。在求解微分方程的逆谱分析问题时，这些迹线也很有用。