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Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization
Authors:Sergio Albeverio  Jose Manuel Bayod  Cristina Perez-Garcia  Roberto Cianci  Andrew Khrennikov
Affiliation:(1) Institute of Applied Mathematics, Bonn University, 53013 Bonn, Germany;(2) Departamento de Matematicas, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain;(3) Dipartimento di Matematica, Universita di Genova, Italy;(4) Department of Mathematics, University of Växjö, 35195 Växjö, Sweden
Abstract:We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).
Keywords:non-Archimedean Hilbert space  p-adic quantization  precision of a measurement  symmetric and orthogonal operators  isometric orthogonal operators  Cauchy–  Buniakovski–  Schwarz inequality  majorant and self-polar norms  p-adic Gaussian distribution  p-adic analiticity
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