Relativistic Wave Equations on the Lattice: An Operational Perspective

Faustino, Nelson

doi:10.1007/978-3-030-23854-4_21

Cited by 4 publications

(27 citation statements)

References 27 publications

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“…As studied in depth by the author in a previous study, the above construction yields as a natural exploitation of an abstract framework involving manipulations of shift‐invariant operators that admit formal series expansions in terms of the time derivative ∂ t as follows:

L_{t} = \sum_{k = 1}^{\infty} b_{k} \frac{{(\partial_{t})}^{k}}{k!}, with b_{k} = {[{(L_{t})}^{k} t^{k}]}_{t = 0} .

Such construction looks similar to Constales and De Ridder's work (cf Faustino, Remark 4.1), except that the − i L t operator (

i = \sqrt{- 1}

)–appearing quite often on Dirac's equation (cf ,. subsection 5.…”

Section: Problem Setupmentioning

confidence: 63%

“…Here, we recall that the left‐hand side of admits the formal series expansion (cf Faustino, Example 2.3.)

Ψ (x, t + τ) - Ψ (x, t - τ) = 2 \sinh (τ \partial_{t}) Ψ (x, t) .

Similarly to Faustino,, subsection 3.2. one can conclude from the identity that the solution of the Cauchy problem may be represented through the exponential generating function (EGF) type expansion

normalΨ false(x, t false) = true \sum_{k = 0}^{\infty} \frac{G_{k} false(t; - τ, 2 τ false)}{k!} {((), 2 τ e_{0} D_{h} + τ^{2} e_{2 n + 1} e_{0} Δ_{h})}^{k} Φ_{0} false(x false) .

…”

Section: The Discrete Cauchy‐kovalevskaya Approach Explainedmentioning

confidence: 80%

“…Here,

Δ_{h} normalΨ false(x, t false) = true \sum_{j = 1}^{n} \frac{normalΨ false(x + h e_{j}, t false) + normalΨ false(x - h e_{j}, t false) - 2 normalΨ false(x, t false)}{h^{2}}

corresponds to the finite difference action of the discrete Laplacian on the space‐time lattice

h Z^{n} \times τ Z_{\geq 0}

(cf Faustino, Subsection 1.2. ), whereas L t is a finite difference operator satisfying the second‐order constraint

L_{t} (L_{t} Ψ (x, t)) = \frac{Ψ (x, t + τ) + Ψ (x, t - τ) - 2 Ψ (x, t)}{τ^{2}} .

For the case where L t is a finite difference operator defined as

L_{t} Ψ (x, t) = \frac{Ψ (x, t + \frac{τ}{2}) - Ψ (x, t - \frac{τ}{2})}{τ},

it was depicted by the author in his recent contribution (cf Faustino, Subsection 4.1.) that the solution of the difference‐difference evolution problem may be derived from the Cauchy principal values representations for the Chebyshev polynomials of first and second kind, T k ( λ ) resp.…”

Section: Problem Setupmentioning

confidence: 99%

“…By the substitution of into we recognize, after some umbral calculus manipulations involving the inverse

L^{- 1} false(s false) = \frac{1}{τ} \sinh^{- 1} ((), \frac{s}{2})

of the function

L false(s false) = 2 \sinh false(τ s false)

(cf Faustino, Appendix A) that

\begin{matrix} right Ψ (x, t) & left = \sum_{m = 0}^{\infty} \frac{2^{2 m} G_{2 m} (t; - τ, 2 τ)}{(2 m)!} {(- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2})}^{m} {normalΦ}_{0} (x) + & right \\ right & left + \sum_{m = 0}^{\infty} \frac{2^{2 m + 1} G_{2 m + 1} (t; - τ, 2 τ)}{(2 m + 1)!} {(- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2})}^{m} [τ {bolde}_{0} D_{h} {normalΦ}_{0} (x) + \frac{τ^{2}}{2} {bolde}_{2 n + 1} {bolde}_{0} {normalΔ}_{h} {normalΦ}_{0} (x)] \\ right & left = \cosh (\frac{t}{τ} \sinh^{- 1} (\sqrt{- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2}})) normal... \end{matrix}

…”

Section: The Discrete Cauchy‐kovalevskaya Approach Explainedmentioning

confidence: 99%

“…In the present paper, we reformulate the approaches considered on both papers. The construction stated in Section makes use of Roman's umbral calculus approach and of the discrete Fourier theory considered by Gürlebeck and Sprößig on their former book, following up author's recent contribution . In Section , it was obtained an explicit space‐time integral representation for the Cauchy‐Kovalevskaya extension on the momentum space, based on Cauchy principal value representations for , and on a Laplace transform identity associated to the generalized Mittag‐Leffler function (cf Samko et al, p. 21)

E_{α, β} false(z false) = true \sum_{k = 0}^{\infty} \frac{z^{k}}{normalΓ false(β + α k false)}, for ℜ false(α false) > 0 & ℜ false(β false) > 0 .

…”

Section: Introductionmentioning

confidence: 99%

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A note on the discrete Cauchy‐Kovalevskaya extension

Faustino

2018

Math Methods in App Sciences

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In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.

show abstract

L_{t} = \sum_{k = 1}^{\infty} b_{k} \frac{{(\partial_{t})}^{k}}{k!}, with b_{k} = {[{(L_{t})}^{k} t^{k}]}_{t = 0} .

Such construction looks similar to Constales and De Ridder's work (cf Faustino, Remark 4.1), except that the − i L t operator (

i = \sqrt{- 1}

)–appearing quite often on Dirac's equation (cf ,. subsection 5.…”

Section: Problem Setupmentioning

confidence: 63%

“…Here, we recall that the left‐hand side of admits the formal series expansion (cf Faustino, Example 2.3.)

Ψ (x, t + τ) - Ψ (x, t - τ) = 2 \sinh (τ \partial_{t}) Ψ (x, t) .

Similarly to Faustino,, subsection 3.2. one can conclude from the identity that the solution of the Cauchy problem may be represented through the exponential generating function (EGF) type expansion

normalΨ false(x, t false) = true \sum_{k = 0}^{\infty} \frac{G_{k} false(t; - τ, 2 τ false)}{k!} {((), 2 τ e_{0} D_{h} + τ^{2} e_{2 n + 1} e_{0} Δ_{h})}^{k} Φ_{0} false(x false) .

…”

Section: The Discrete Cauchy‐kovalevskaya Approach Explainedmentioning

confidence: 80%

“…Here,

Δ_{h} normalΨ false(x, t false) = true \sum_{j = 1}^{n} \frac{normalΨ false(x + h e_{j}, t false) + normalΨ false(x - h e_{j}, t false) - 2 normalΨ false(x, t false)}{h^{2}}

corresponds to the finite difference action of the discrete Laplacian on the space‐time lattice

h Z^{n} \times τ Z_{\geq 0}

(cf Faustino, Subsection 1.2. ), whereas L t is a finite difference operator satisfying the second‐order constraint

L_{t} (L_{t} Ψ (x, t)) = \frac{Ψ (x, t + τ) + Ψ (x, t - τ) - 2 Ψ (x, t)}{τ^{2}} .

For the case where L t is a finite difference operator defined as

L_{t} Ψ (x, t) = \frac{Ψ (x, t + \frac{τ}{2}) - Ψ (x, t - \frac{τ}{2})}{τ},

Section: Problem Setupmentioning

confidence: 99%

“…By the substitution of into we recognize, after some umbral calculus manipulations involving the inverse

L^{- 1} false(s false) = \frac{1}{τ} \sinh^{- 1} ((), \frac{s}{2})

of the function

L false(s false) = 2 \sinh false(τ s false)

(cf Faustino, Appendix A) that

\begin{matrix} right Ψ (x, t) & left = \sum_{m = 0}^{\infty} \frac{2^{2 m} G_{2 m} (t; - τ, 2 τ)}{(2 m)!} {(- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2})}^{m} {normalΦ}_{0} (x) + & right \\ right & left + \sum_{m = 0}^{\infty} \frac{2^{2 m + 1} G_{2 m + 1} (t; - τ, 2 τ)}{(2 m + 1)!} {(- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2})}^{m} [τ {bolde}_{0} D_{h} {normalΦ}_{0} (x) + \frac{τ^{2}}{2} {bolde}_{2 n + 1} {bolde}_{0} {normalΔ}_{h} {normalΦ}_{0} (x)] \\ right & left = \cosh (\frac{t}{τ} \sinh^{- 1} (\sqrt{- τ^{2} {normalΔ}_{h} + \frac{τ^{4}}{4} {normalΔ}_{h}^{2}})) normal... \end{matrix}

…”

Section: The Discrete Cauchy‐kovalevskaya Approach Explainedmentioning

confidence: 99%

E_{α, β} false(z false) = true \sum_{k = 0}^{\infty} \frac{z^{k}}{normalΓ false(β + α k false)}, for ℜ false(α false) > 0 & ℜ false(β false) > 0 .

…”

Section: Introductionmentioning

confidence: 99%

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A note on the discrete Cauchy‐Kovalevskaya extension

Faustino

2018

Math Methods in App Sciences

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On fractional semidiscrete Dirac operators of Lévy–Leblond type

Faustino

2023

Mathematische Nachrichten

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In this paper, we introduce a wide class of space‐fractional and time‐fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space‐time lattice hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty )$ (h>0$h>0$), resembling to fractional semidiscrete counterparts of the so‐called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup {}exp(−teiθ(−normalΔh)α)t≥0$\left\lbrace \exp (-te^{i\theta }(-\Delta _h)^{\alpha })\right\rbrace _{t\ge 0}$, carrying the parameter constraints 0<α≤1$0<\alpha \le 1$ and |θ|≤απ2$|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty )$.

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Discrete pseudo‐differential operators in hypercomplex analysis

Cerejeiras

Kähler

Lucas

2021

Math Methods in App Sciences

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In this paper we discuss discrete pseudo‐differential operators in the context of hypercomplex analysis. We will provide the definitions and establish the necessary calculus which will be used to discuss questions like boundedness and compactness of operators. In the end, we will discuss well‐known examples from hypercomplex analysis as well as give new examples including discrete Dirac operators with non‐constant coefficients.

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Relativistic Wave Equations on the Lattice: An Operational Perspective

Cited by 4 publications

References 27 publications

A note on the discrete Cauchy‐Kovalevskaya extension

A note on the discrete Cauchy‐Kovalevskaya extension

On fractional semidiscrete Dirac operators of Lévy–Leblond type

Discrete pseudo‐differential operators in hypercomplex analysis

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