A theorem of Strichartz states that if a uniformly bounded bi-infinite sequence of functions on Euclidean spaces satisfies the property that the Laplacian acting on one function in the sequence yields the next, then every function in this sequence is an eigenfunction of the Laplacian. This result was later extended by replacing the standard Euclidean Laplacian with operators such as the d’Alembertian, the harmonic oscillator, and constant-coefficient linear partial differential operators on $\mathbb{R}^n$.

In this talk, we will explore several variants of this result for homogeneous trees, where the Euclidean Laplacian is replaced by the combinatorial Laplacian and the uniform boundedness condition is appropriately adjusted. We will then explore possible generalizations when the combinatorial Laplacian is substituted with multipliers on homogeneous trees. After presenting the result in this broader context, we will narrow our focus to specific cases, including key examples of multiplier operators such as the heat and Schrödinger operators, as well as ball and sphere averages of functions. The talk is based on a joint work with Rudra P. Sarkar.

The video of this talk is available on the IISc Math Department channel.