caa a22 4500 475839633 CHVBK 20180406123858.0 cr unu---uuuuu 170329e20001001xx s 000 0 eng 10.1023/A:1006457731833 doi (NATIONALLICENCE)springer-10.1023/A:1006457731833 Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups [Elektronische Daten] [S. Albeverio, E. Gordon, A. Khrennikov] A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R n , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621-648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used. Kluwer Academic Publishers, 2000 finite-dimensional approximations nationallicence locally compact Abelian groups nationallicence Schrödinger operators nationallicence p-adic numbers nationallicence nonstandard analysis nationallicence Albeverio S. Institut fuer Angewandte Mathematik Abtlg. Stochastik, Rheinische Friedrich-Wilhelms-Universitaet Bonn, Wegelerstrasse 6, 53115, Bonn, Germany. e-mail aut Gordon E. Department Mathematics, University of Illinois, 1409 West Green Street, 61801, Urbana, IL, U.S.A. aut Khrennikov A. Department of Mathematics, Statistics and Computer Science, University of Väjö, S 35 195, Sweden aut Acta Applicandae Mathematica Kluwer Academic Publishers 64/1(2000-10-01), 33-73 0167-8019 64:1<33 2000 64 10440 https://doi.org/10.1023/A:1006457731833 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1006457731833 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Albeverio S. Institut fuer Angewandte Mathematik Abtlg. Stochastik, Rheinische Friedrich-Wilhelms-Universitaet Bonn, Wegelerstrasse 6, 53115, Bonn, Germany. e-mail aut NATIONALLICENCE 700 1- Gordon E. Department Mathematics, University of Illinois, 1409 West Green Street, 61801, Urbana, IL, U.S.A aut NATIONALLICENCE 700 1- Khrennikov A. Department of Mathematics, Statistics and Computer Science, University of Väjö, S 35 195, Sweden aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 64/1(2000-10-01), 33-73 0167-8019 64:1<33 2000 64 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer