We examine an example of a self-adjoint ordinary differential operator going back to Naimark. This operator is remarkable because its point and continuous spectra intersect. We find the spectrum and eigenfunctions of this operator “following Landau and Lifshitz,” i.e., following the rules stated in their book Quantum Mechanics and based on plausible heuristic physical arguments and analogies with linear algebra, which, to our knowledge, have not been rigorously mathematically justified so far. Then we adduce arguments in support of reasonableness of the results obtained by this method, which is conventional for physicists. The arguments are based on the analysis of the independently calculated Green function of the operator.

中文翻译：

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自伴微分算子“跟随Landau和Lifshitz”的谱分析通过绿色函数的验证

我们研究一个可以追溯到Naimark的自伴普通微分算子的示例。该算子之所以出色，是因为其点和连续光谱相交。我们发现该算子的谱和本征函数“遵循Landau和Lifshitz”，即遵循他们的《量子力学》一书中所述的规则，并基于合理的启发式物理论证和与线性代数的类比，据我们所知，这并不是严格的到目前为止在数学上是合理的。然后，我们提出论点以支持这种方法获得的结果的合理性，这对于物理学家是常规的。参数基于对运算符的独立计算的格林函数的分析。