scholarly journals Transformations on white noise functions associated with second order differential operators of diagonal type

1998 ◽  
Vol 149 ◽  
pp. 173-192 ◽  
Author(s):  
Dong Myung Chung ◽  
Un Cig Ji ◽  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Differential Operators ◽  
Transformation Groups ◽  
Cauchy Problems ◽  
Noise Distribution ◽  
Number Operator ◽  
Parameter Transformation ◽  
Infinite Dimensional ◽  
White Noise Distribution Theory ◽  
White Noise Distribution

Abstract.A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

POISSON–MASLOV TYPE FORMULAS FOR SCHRÖDINGER EQUATIONS WITH MATRIX-VALUED POTENTIALS

2007 ◽  
Vol 10 (04) ◽  
pp. 641-649 ◽  
Author(s):  
N. N. SHAMAROV
Keyword(s):  
Momentum Space ◽  
Cauchy Data ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Cauchy Problems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Continuous Mappings ◽  
Piecewise Continuous ◽  
Transition Amplitudes

Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved under following assumptions:. — equations are written in momentum form;. — the potentials are Fourier transformed matrix-valued measures with, in general, noncommuting values;. — initial Cauchy data are good enough. The solutions at time t are presented in form of integrals over some spaces of piecewise continuous mappings of the segment [0, t] to a finite-dimensional momentum space, and measures of the integration are countably additive but matrix-valued (resulting in matrices of ordinary Lebesgue integrals). Known results gave solutions either when the measure had commuting (complex) values, or when the integration over infinite dimensional spaces was quite symbolic, or when such integrating was of chronological type and hence more complicated. The method used below is based on technique of matrix-valued transition amplitudes.

scholarly journals DENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS

2020 ◽  
Vol 8 (2) ◽  
pp. 122-126
Author(s):  
V. Slyusarchuk
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Cauchy Problems ◽  
Number Of Solutions ◽  
Computational Mathematics ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Cauchy Problem ◽  
Solutions Of Equations

When finding solutions of differential equations it is necessary to take into account the theorems on innovation and unity of solutions of equations. In case of non-fulfillment of the conditions of these theorems, the methods of finding solutions of the studied equations used in computational mathematics may give erroneous results. It should also be borne in mind that the Cauchy problem for differential equations may have no solutions or have an infinite number of solutions. The author presents two statements obtained by the author about the denseness of sets of the Cauchy problem without solutions (in the case of infinite-dimensional Banach space) and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchy problems. Using two examples of the Cauchy problem for differential equations, the imperfection of some methods of computational mathematics for finding solutions of the studied equations is shown.

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