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We show that the Schr¨odinger operator associated with a physical system over a local field can be approximated in a very strong sense by finite Schr¨odinger operators. Some striking numerical results are included at the end of the article.

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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.2010 Mathematics Subject Classification. 81Q65, 60B10, 47G30, 41A99.