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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-\Delta _{\mathcal {S}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi mathvariant="script">S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be the Laplace operator in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{S} \subset \mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> on a waveguide shaped surfaces, i.e., <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathcal {S}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(-\Delta _{\mathcal {S}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi mathvariant="script">S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.</p>

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Spectrum of the Laplacian in waveguide shaped surfaces

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Diana C. S. Bello

摘要

Let \(-\Delta _{\mathcal {S}}\) - Δ S be the Laplace operator in \(\mathcal{S} \subset \mathbb {R}^3\) S ⊂ R 3 on a waveguide shaped surfaces, i.e., \({\mathcal {S}}\) S is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of \(-\Delta _{\mathcal {S}}\) - Δ S and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.