<p>In this paper we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1164_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\hbar \mathbb {Z}^{n}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ħ</mi> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Contrary to the Euclidean case that was considered in [<CitationRef CitationID="CR2">2</CitationRef>], the discrete fractional Klein-Gordon equation is well-posed in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1164_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell ^{2}\left( \hbar \mathbb {Z}^{n}\right) .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mfenced close=")" open="("> <mi>ħ</mi> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1164_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\hbar \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ħ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Discrete Time-dependent wave equations II. Semiclassical Fractional Klein-Gordon equation

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发表日期：

2025年5月24日

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Aparajita Dasgupta,
Michael Ruzhansky,
Abhilash Tushir

摘要

In this paper we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice $\hbar \mathbb {Z}^{n}.$ $ħ Z^{n} .$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell ^{2}\left( \hbar \mathbb {Z}^{n}\right) .$ $ℓ^{2} (ħ, Z^{n}) .$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $\hbar \rightarrow 0$ $ħ \to 0$ .