Feynman and quasi-Feynman formulas for evolution equations

Remizov, Ivan D.

doi:10.1134/s1064562417050052

Cited by 10 publications

(8 citation statements)

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“…The Chernoff theorem [17,14] allows us to reduce the problem of finding e tH to the problem of finding an appropriate operator-valued function S(t) = I + tH + o(t), which is called the Chernoff function, and then use the Chernoff formula e tH = lim n→∞ S(t/n) n . One advantage of that step is that we can define S(t) by an explicit formula that depends on the coefficients of the operator H. Another advantage is that for each t the operator S(t) is a linear bounded operator, which allows us to define analytic functions of argument S(t) via power series (see examples [40,39,37,41,36]) to obtain a semigroup e −itH t≥0 that solves Schrödinger equation with Hamiltonian H. This idea was introduced in [40] where we defined R(t) = exp − i(S(t) − I) and proved that e −itH = lim n→∞ R(t/n) n . Members of O.G.Smolyanov's group employed Chernoff's theorem using integral operators as Chernoff functions to find solutions to parabolic equations in many cases during the last 15 years: see the pioneering papers [46,45], overview [44], several examples [43,11,38,16,47,42,12] to see the diversity of applications, and recent papers [39,13,41,9,5,29,8,36,27,22,24,48,21,7,35].…”

Section: State Of the Artmentioning

confidence: 99%

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

Remizov

2019

Journal of Mathematical Physics

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We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C 0 -semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth and satisfy some other conditions then there exists a solution-giving C 0 -semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e. as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.

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Section: State Of the Artmentioning

confidence: 99%

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

Remizov

2019

Journal of Mathematical Physics

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“…In both cases the potential may be unbounded which covers the Hamiltonian of quantum (an)harmonic oscillator, this was not done in [50]. See also [62] for short introduction to quasi-Feynman formulas and the calculus of Chernoff functions.…”

Section: Feynman Formulas and Quasi-feynman Formulasmentioning

confidence: 99%

“…Here we assume F = L 2 (R d ), L = iH, u 0 = ψ 0 and u(t) = ψ(t). A representation of the function ψ in this form is a quasi-Feynman formula with generalized functions (=distributiuons) under the integral sign [62]. See also discussion in the end of section 3.3.…”

Section: Proofmentioning

confidence: 99%

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

Remizov

2018

Potential Anal

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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.

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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

Preprint

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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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Feynman and quasi-Feynman formulas for evolution equations

Cited by 10 publications

References 9 publications

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

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

Speed of convergence of Chernoff approximations to solutions of evolution equations

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