Abstract

We study the spectral properties of the Cauchy problem for the differential operator \(-u^{{\prime \prime }}(x)+\alpha u^{{\prime \prime }}(-x) \) with an involution for \(\alpha \) satisfying the inequalities \(0<|\alpha |<1 \). Based on the analysis of the spectrum and the Green’s function constructed here, it is shown that if the parameter \(\varkappa =\sqrt {(1-\alpha )/(1+\alpha )}\) is irrational, then the system of root functions is complete but is not a basis in \(L_2\). In the opposite case, it is established that the root functions can be chosen to form an unconditional basis in \(L_2 \).

中文翻译：

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二阶对合算子柯西问题的谱性质

摘要

我们研究了Cauchy问题的光谱特性的微分算子 \（ - U ^ {{\素\素}}（X）+ \阿尔法Ú^ {{\素\素}}（ - X）\）与满足不等式\（0 <| \ alpha | <1 \）的\（\ alpha \）的对合。根据频谱分析和此处构造的格林函数，可以看出，如果参数\（\ varkappa = \ sqrt {（1- \ alpha /（1+ \ alpha}} \）是不合理的，则根函数系统完整，但不是\（L_2 \）的基础。在相反的情况下，确定可以选择根函数以形成\（L_2 \）中的无条件基础 。