We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions, such as asymptotics or other analytic properties. Instead, we carefully sample the range of the resolvent operator to construct smooth and accurate wave packet approximations to generalized eigenfunctions. We prove high-order convergence in key topologies, including weak-star convergence for distributional eigenfunctions, uniform convergence on compact sets for locally smooth generalized eigenfunctions, and convergence in seminorms for separable Frechet spaces, covering the majority of physical scenarios. The method's performance is illustrated with applications to both differential and integral operators, culminating in the computation of spectral measures and generalized eigenfunctions for an operator associated with Poincare's internal waves problem. These computations corroborate experimental results and highlight the method's utility for a broad range of spectral problems in physics.

翻译：我们提出了一种简单、通用且收敛的方案，用于计算装配希尔伯特空间上具有连续谱的自伴算子的广义特征函数。我们的方法不需要预先了解特征函数的渐近行为或其他解析性质。相反，我们通过对预解算子的值域进行精细采样，来构造广义特征函数的平滑且精确的波包近似。我们证明了该方法在关键拓扑下的高阶收敛性，包括分布型特征函数的弱星收敛、局部光滑广义特征函数在紧集上的一致收敛，以及可分离弗雷歇空间中的半范数收敛，这涵盖了大多数物理场景。通过应用于微分算子和积分算子的实例，展示了该方法的性能，最终计算了与庞加莱内波问题相关算子的谱测度和广义特征函数。这些计算验证了实验结果，并凸显了该方法在物理学广泛谱问题中的实用性。