Feynman's formula in phase space for systems of pseudodifferential equations with analytic symbols

Egikyan, R. S.; Ktitarev, D. V.

doi:10.1007/bf01262176

Cited by 2 publications

(3 citation statements)

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“…The second operator is bounded in the L 2 (R d )-norm, and 10 the first is closable and its closure is a generator of a unitary one-paremeter semigroup (OPS) in L 2 (R d ); hence 10 the sum A + B generates an OPS which is the exponent resolving (2), and moreover ϕ(t, ·) = e t(A+B) ϕ 0 = lim n→∞ (e tA/n e tB/n ) n ϕ 0 in L 2 (R d ) (Trotter formula). Now we will give a representation of e τ (A+B) ϕ 0 in terms of the measure M .…”

Section: Solution To Finite Dimensional Schrödinger-cauchy Problem Wimentioning

confidence: 99%

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Poisson–maslov Type Formulas for Schrödinger Equations With Matrix-Valued Potentials

Shamarov

2007

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

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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.

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Section: Solution To Finite Dimensional Schrödinger-cauchy Problem Wimentioning

confidence: 99%

“…1 and also in Ref. 2 (where the ideas of the book 1 are exploited) some Dyson type series are used. In contrast, in this paper we follow the quite different approach of Smolyanov 3 who used some Trotter type products formulas.…”

Section: Introductionmentioning

confidence: 99%

Poisson–maslov Type Formulas for Schrödinger Equations With Matrix-Valued Potentials

Shamarov

2007

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

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“…In [5], uniqueness of solutions is not claimed; in [11] and [12], functional integrals are understood only as generalized integrals; in [13] (and in earlier articles by Ktitarev), the authors proceed under assumptions that are more restrictive when applied to our situation. This method can also be adapted as a generalization, to the case of infinite-dimensional algebras, of values of the coefficients (generally speaking, they depend on the "space" variables) occurring on the right-hand side of the evolution equation; in particular, for the second-order super-differential equations presented in [5], the results of which do not overlap those described below, and in which, in order to construct a functional Poisson distribution analogous to the measure M 0 , the authors use not finite-dimensional distributions (= Trotter approximations) but the Dyson series analogous to that used in [1] to solve the Schrödinger equation.…”

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confidence: 99%

A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

Shamarov¹

2005

Trans. Moscow Math. Soc.

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Abstract. A family {M p } p∈R d of cylindrical measures is constructed on the space of functions ( = trajectories) f : [0, +∞) → R d such that for every t ≥ 0 the formularepresents a solution of the Cauchy problem(with respect to the required function ψ : The integral kernels ("Green's functions") of the corresponding solution operators, which can be approximated (using Trotter's formula) by integrals of finite multiplicity of the expressions explicitly defined by the ingredients of the original equation, are (matrix-valued) transition measures that give cylindrical measures M p similarly to the way Markov transition probabilities give the distribution of a Markov process.A new method using matrix-valued transition measures is applied in this paper, leading to a stronger result for the Dirac equation than those that follow under assumptions analogous to ours from results in previous papers. In [5], uniqueness of solutions is not claimed; in [11] and [12], functional integrals are understood only as generalized integrals; in [13] (and in earlier articles by Ktitarev), the authors proceed under assumptions that are more restrictive when applied to our situation. This method can also be adapted as a generalization, to the case of infinite-dimensional algebras, of values of the coefficients (generally speaking, they depend on the "space" variables) occurring on the right-hand side of the evolution equation; in particular, for the second-order super-differential equations presented in [5], the results of which do not overlap those described below, and in which, in order to construct a functional Poisson distribution analogous to the measure M 0 , the authors use not finite-dimensional distributions (= Trotter approximations) but the Dyson series analogous to that used in [1] to solve the Schrödinger equation. Our method goes back to the similarity indicated in [14] between the properties of complex

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Feynman's formula in phase space for systems of pseudodifferential equations with analytic symbols

Cited by 2 publications

References 6 publications

Poisson–maslov Type Formulas for Schrödinger Equations With Matrix-Valued Potentials

Poisson–maslov Type Formulas for Schrödinger Equations With Matrix-Valued Potentials

A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

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