On the Fourier transform and the spectral properties of the \(p\)-adic momentum and Schrödinger operators.

*(English)*Zbl 0927.46060The authors continue the study of the \(p\)-adic model of quantum mechanics initiated in their previous papers [S. Albeverio, A. Khrennikov, J. Phys. A, Math. Gen. 29, 5515-5527 (1996; Zbl 0903.46073); S. Albeverio, R. Cianci, A. Khrennikov, J. Phys. A, Math. Gen. 30, 881-889 (1997)]. In particular, a version of the Fourier transform is introduced, which is adapted to the spaces used in the paper, \(L_2\)-spaces with respect to the \(p\)-adic Gaussian distribution.

In the momentum representation the analogue of the momentum operator is similar to the position operator of the coordinate representation, which was studied earlier. As a result, spectral properties of the momentum operator are investigated. A possible physical meaning of the results is discussed, in particular the idea of a \(p\)-adic description of the finite exactness of measurement.

For another recent \(p\)-adic model of quantum mechanics, which is closely related to classical constructions of \(p\)-adic analysis, see the paper by the reviewer [A. N. Kochubei, J. Phys. A, Math. Gen. 29, 6375-6378 (1996; Zbl 0905.46051)].

In the momentum representation the analogue of the momentum operator is similar to the position operator of the coordinate representation, which was studied earlier. As a result, spectral properties of the momentum operator are investigated. A possible physical meaning of the results is discussed, in particular the idea of a \(p\)-adic description of the finite exactness of measurement.

For another recent \(p\)-adic model of quantum mechanics, which is closely related to classical constructions of \(p\)-adic analysis, see the paper by the reviewer [A. N. Kochubei, J. Phys. A, Math. Gen. 29, 6375-6378 (1996; Zbl 0905.46051)].

Reviewer: Anatoly N.Kochubei (Kiev)

##### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

81S05 | Commutation relations and statistics as related to quantum mechanics (general) |

81P15 | Quantum measurement theory, state operations, state preparations |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |