Abstract

For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral asymptotics and the formula for the regularized trace look the same as for the classical Sturm–Liouville operator on a segment with smooth potential. Further, it is shown that in the case of nontrivial monodromy the spectral asymptotics substantially depends on the commensurability of the parts into which the segment is partitioned by the singular point \(\theta \): if \(\theta \) is rational, then the spectrum is divided into finitely many series, each going to infinity along “its own” parabola. In this case, the regularized trace formula is significantly more complicated and does not show any similarity with the classical formula.

中文翻译：

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在弦上具有规则奇点的曲线上的Sturm–Liouville算子的正则迹线

摘要

对于分段平滑曲线上的Sturm–Liouville算子，我们研究了连接该曲线端点的线段上的势不可积分奇异谱对算子渐近性的影响。结果表明，在奇异点不引起邻域解分支的情况下（平凡单峰情况），频谱渐近性和正则化迹线的公式与经典Sturm相同。 Liouville算子在具有平稳潜力的线段上。此外，还表明，在非平凡单峰情况下，频谱渐近性基本上取决于该段被奇点\（\ theta \）划分的各部分的可比性 ：if \（\ theta \）如果是有理数，则将频谱分成有限的多个系列，每个系列沿着“其自己的”抛物线到达无穷大。在这种情况下，正规化的跟踪公式要复杂得多，并且与经典公式没有任何相似之处。