Lyalinov Mikhail A
Dept. Mathematics and Mathematical Physics, Saint-Petersburg University, Universitetskaya nab. 7/9, 199034, S.-Petersburg, Russia.
Proc Math Phys Eng Sci. 2020 Sep;476(2241):20200179. doi: 10.1098/rspa.2020.0179. Epub 2020 Sep 16.
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).
研究了具有奇异势的拉普拉斯算子的本征函数及其在大距离处的渐近行为，该奇异势的支撑集位于三维空间中的圆形圆锥面上。在不完全变量分离的框架下，根据具有亚纯势的辅助泛函差分方程的解，得到了本征函数的康托罗维奇 - 列别杰夫（KL）型积分表示。通过将泛函差分方程化为具有有界自伴积分算子的积分方程来研究其解。为了计算本征函数渐近性的主导项，将KL积分表示转换为适合鞍点技术应用的索末菲尔德型积分。在所谓奇异方向的小角度邻域之外，渐近表达式具有距离上指数衰减的基本形式。然而，在奇异方向的渐近小邻域内，渐近的主导项还依赖于与抛物柱面函数（韦伯函数）密切相关的特殊函数。
