Quasi-Feynman Formulas providing Solutions of Multidimensional Schrödinger Equations with Unbounded Potential

Remizov, Ivan D.; Starodubtseva, M. F.

doi:10.1134/s0001434618110214

Cited by 9 publications

(5 citation statements)

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“…Case 3: shifts and averaging. One more approach to construct Chernoff approximations for semigroups and Schrödinger groups generated by differential and pseudo-differential operators is based on shift operators (see [63,64]), averaging (see [10,57]) and their combination (see [10,9]). Let us demonstrate this method by means of simplest examples.…”

Section: 2mentioning

confidence: 99%

“…Hence the family (S t ) t≥0 is Chernoff equivalent to the heat semigroup (1) on X. Extending (S t ) t≥0 to the d−dimensional case and applying the "rotation" techniques in X = L 2 (R d ), one obtains Chernoff approximation for the Schrödinger group (e it∆ ) t≥0 ( [64]). Further, one can apply the techniques of Sections 2.1-2.5, to construct Chernoff approximations for Schrödinger groups generated by more complicated differential and pseudo-differential operators.…”

Section: 2mentioning

confidence: 99%

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The method of Chernoff approximation

Butko

2019

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This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes, numerical schemes for PDEs and SDEs, path integrals. We discuss Chernoff approximations for operator semigroups and Schrödinger groups. In particular, we consider Feller semigroups in R d , (semi)groups obtained from some original (semi)groups by different procedures: additive perturbations of generators, multiplicative perturbations of generators (which sometimes corresponds to a random time-change of related stochastic processes), subordination of semigroups / processes, imposing boundary / external conditions (e.g., Dirichlet or Robin conditions), averaging of generators, "rotation" of semigroups. The developed techniques can be combined to approximate (semi)groups obtained via several iterative procedures listed above. Moreover, this method can be implemented to obtain approximations for solutions of some time-fractional evolution equations, although these solutions do not posess the semigroup property.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

The method of Chernoff approximation

Butko

2019

Preprint

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“…Brief history and overview of the results obtained up to 2017 in constructing Chernoff approximations of e tL for several classes of operators L can be found in [3]. Several papers on the topic showing the diversity of cases studied are [14,15,16,17,18,19,20,21], see also [12,22]. Speed of convergence of Chernoff approximations were studied in [23,13,8,9,7].…”

Section: Introductionmentioning

confidence: 99%

Chernoff approximations as a way of finding the resolvent operator with applications to finding the solution of linear ODE with variable coefficients

Remizov¹

2023

Preprint

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The method of Chernoff approximation is a powerful and flexible tool of functional analysis that in many cases allows expressing exp(tL) in terms of variable coefficients of linear differential operator L. In this paper we prove a theorem that allows us to apply this method to find the resolvent of operator L. We demonstrate this on the second order differential operator. As a corollary, we obtain a new representation of the solution of an inhomogeneous second order linear ordinary differential equation in terms of functions that are the coefficients of this equation playing the role of parameters for the problem.

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“…Remark 8. The discussion of the above-mentioned model examples rises a hope for creation of universal methods of construction of fast-converging Chernoff approximations (in particular, Feynman formulas [8] and their analogues [9,10]) for evolution equations with variable coefficients.…”

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confidence: 99%

Speed of convergence of Chernoff approximations to solutions of evolution equations

Vedenin

Voevodkin

Galkin

et al. 2019

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This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.Introduction. Since the middle of the XX century it is a well known fact [1,2] that the solution of a well-posed Cauchy problem for a linear evolution partial differential equation (examples: Schödinger equation, parabolic equations) is given by a strongly continuous semigroup of linear bounded operators whose infinitesimal generator is a (usually unbounded) linear operator from the right-hand side of the evolution equation. Let us explain this in more details and introduce some notation which will be useful for the main text.Let X be an infinite set, and F be a Banach space of (not necessarily all) number-valued functions on X, and let L be a closed linear operator L : Dom(L) → F with the domain Dom(L) ⊂ F dense in F . We consider the Cauchy problem for the evolution equation

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Quasi-Feynman Formulas providing Solutions of Multidimensional Schrödinger Equations with Unbounded Potential

Cited by 9 publications

References 6 publications

The method of Chernoff approximation

The method of Chernoff approximation

Chernoff approximations as a way of finding the resolvent operator with applications to finding the solution of linear ODE with variable coefficients

Speed of convergence of Chernoff approximations to solutions of evolution equations

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