Since their introduction, generalized Toeplitz structures (in the sense of L. Boutet de Monvel and V. Guillemin) over contact manifolds have found numerous applications in fields such as CR geometry, analysis, and quantization. More recently, S. Zelditch introduced the concept of dynamical Toeplitz operators to study the dynamics of quantized contact transformations. These operators, closely tied to the geometry of the underlying manifold, have demonstrated significant applications in both geometry and analysis.

The aim of this talk is to provide a simple introduction to the geometric and analytical context of these operators on Grauert tube boundaries and to present, as an application, their connection to the complexification of Laplacian eigenfunctions.