This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter. Consideration of this type of equations is motivated by applications in diffraction theory and by construction of eigenfunctions for the Laplace operator in angular domains. In particular, such eigenfunctions describe eigenoscillations of acoustic waves in angular domains with ‘semitransparent’ boundary conditions. For negative values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The obtained results are applied for studying the behaviour of eigenfunctions for the Laplace operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary. At infinity, the eigenfunctions vanish exponentially as was expected. However, the rate of such decay depends on the observation direction. In particular, in a vicinity of some directions, the regime of decay is switched from one to another and such asymptotic behaviour is described by a Fresnel-type integral.

这项工作研究了具有属于一类特殊亚纯函数的势的二阶泛函差分方程。方程取决于光谱参数。考虑这种类型的方程是由衍射理论中的应用和角域中拉普拉斯算子的特征函数的构造所激发的。特别是，这种本征函数描述了具有“半透明”边界条件的角域中声波的本征振荡。对于谱参数的负值，我们研究方程的本质和离散谱，并描述相应解的性质。该研究基于将泛函差分方程简化为具有对称核的积分方程。为确保离散谱存在的势制定了充分条件。所得结果用于研究拉普拉斯算子在公共边界上具有 Robin 型边界条件的相邻角域中的特征函数行为。在无穷远处，本征函数按预期呈指数消失。然而，这种衰减率取决于观察方向。特别是，在某些方向附近，衰减状态从一个切换到另一个，这种渐近行为由菲涅耳型积分描述。正如预期的那样，本征函数呈指数级消失。然而，这种衰减率取决于观察方向。特别是，在某些方向附近，衰减状态从一个切换到另一个，这种渐近行为由菲涅耳型积分描述。正如预期的那样，本征函数呈指数级消失。然而，这种衰减率取决于观察方向。特别是，在某些方向附近，衰减状态从一个切换到另一个，这种渐近行为由菲涅耳型积分描述。