Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich–Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).

中文翻译：

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圆锥面上具有δ ′ -相互作用的薛定谔算子的泛函差分方程和本征函数

研究了具有奇异势的拉普拉斯算子的远距离特征函数及其渐近行为，该算子的支撑位于三维空间的圆锥面上。在变量不完全分离的框架内，根据具有亚纯势的辅助函数差分方程的解，获得了本征函数的 Kontorovich-Lebedev (KL) 类型的积分表示。通过将泛函差分方程化简为具有有界自伴随积分算子的积分方程，研究了该方程的解。为了计算本征函数渐近线的前导项，KL积分表示被转换为非常适合鞍点技术应用的Sommerfeld型积分。在所谓的奇异方向的小角度附近之外，渐近表达式采用距离减小的指数的基本形式。然而，在奇异方向的渐近小邻域中，渐近的首项还依赖于一个与抛物柱函数密切相关的特殊函数（韦伯函数）。