Brownian motion and finite approximations of quantum systems over local fields

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman–Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

AN EXAMPLE OF A SUBGROUP SEPARABLE BUT NOT CONJUGACY SEPARABLE GROUP

We give an example of a conjugacy separable, but not subgroup separable group. It is an adaptation of an example of Tavgen and the third author from [Closed orbits and finite approximability with respect to conjugacy of free amalgamated products, Math. Notes58 (1995) 1042–1048] to a theorem of Raptis [On finiteness conditions of certain graphs of groups, Internat. J. Algebra Comput.5 (1995) 719–724].