Quantum mechanics over non-Archimedean number fields

Schrödinger and Bargmann–Fock representations in non-Archimedean quantum mechanics are realized in the spaces $L_2(K^n,dx)$ and $L_2(Z^n,e^{-zz}\,dz\,d\bar z)$ ($K$ is a non-Archimedean field, and $Z=K(\sqrt\tau\,)$ is its quadratic extension) by means of the calculus of pseudodifferential operators.