Feynman Formulas Representation of Semigroups  Generated by Parabolic Difference-Differential Equations

Сакбаев, В. Ж.; Yaakbarieh, A.

doi:10.4236/ajcm.2012.24040

Cited by 2 publications

(2 citation statements)

References 8 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since 2000, O. G. Smolyanov and members of his group succeeded in representing solutions of the Cauchy problem for many evolution equations in form of Feynman formulas (see [39,40,41,42,43,47,48,49,51,52,48,54,61,57,58,44] and refereces therein). The key idea in these representations lies in finding the Chernoff function G for operator L and then applying Chernoff's theorem to obtain the equality e tL u 0 = lim n→∞ G(t/n) n u 0 which apperas to be a Feynman formula, because in all known examples (until [50] was published in 2016, see also [44,63]) G(t) from the equation above was an integral operator, so G(t/n) n was an n-tuple integral operator, giving us a limit of multiple integral where miltiplicity tends to infinity.…”

Section: Feynman Formulas and Quasi-feynman Formulasmentioning

confidence: 99%

Section: Feynman Formulas and Quasi-feynman Formulasmentioning

confidence: 99%

See 1 more Smart Citation

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

Remizov

2018

Potential Anal

View full text Add to dashboard Cite

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

show abstract

Section: Feynman Formulas and Quasi-feynman Formulasmentioning

confidence: 99%

Section: Feynman Formulas and Quasi-feynman Formulasmentioning

confidence: 99%

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

Remizov

2018

Potential Anal

View full text Add to dashboard Cite

show abstract

Explicit formula for evolution semigroup for diffusion in Hilbert space

Remizov

2018

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

View full text Add to dashboard Cite

A parabolic partial differential equation [Formula: see text] is considered, where [Formula: see text] is a linear second-order differential operator with time-independent (but dependent on [Formula: see text]) coefficients. We assume that the spatial coordinate [Formula: see text] belongs to a finite- or infinite-dimensional real separable Hilbert space [Formula: see text]. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator [Formula: see text]. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over [Formula: see text] as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on [Formula: see text]. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in [Formula: see text] is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

show abstract

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

Cited by 2 publications

References 8 publications

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

Explicit formula for evolution semigroup for diffusion in Hilbert space

Contact Info

Product

Resources

About