Analogues of Feynman formulas for ill-posed problems associated with the Schrödinger equation
- Authors: Sakbaev V.G.1, Smolyanov O.G.2
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Affiliations:
- Moscow Institute of Physics and Technology (State University)
- Mechanics and Mathematics Faculty
- Issue: Vol 94, No 3 (2016)
- Pages: 654-658
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/224528
- DOI: https://doi.org/10.1134/S1064562416060132
- ID: 224528
Cite item
Abstract
Representations of Schrödinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrödinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrödinger equation under consideration is ill-posed. From the point of view of the approach of the paper, this means that the problem has no solution in the sense of integral identity for some initial data. The well-posedness of the Cauchy problem can be recovered by extending the operator to a selfadjoint one; however, there exists continuum many such extensions. Feynman iterations whose partial limits are the solutions of all Cauchy problems obtained for various self-adjoint extensions are studied.
About the authors
V. G. Sakbaev
Moscow Institute of Physics and Technology (State University)
Author for correspondence.
Email: fumi2003@mail.ru
Russian Federation, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700
O. G. Smolyanov
Mechanics and Mathematics Faculty
Email: fumi2003@mail.ru
Russian Federation, Moscow, 119991