论文标题
波导中的dirichlet laplacian的频谱与平行横截面
Spectrum of the Dirichlet Laplacian in waveguides with parallel cross-sections
论文作者
论文摘要
令$ω\ subset \ mathbb r^3 $为波导，可以通过沿无界空间曲线在恒定方向上转换横截面获得。考虑$-Δ_Ω在参考曲线的切线向量接收无限时的有限限制的条件下，我们发现$-Δ_Ω^d $的必需频谱。然后，我们陈述了足够的条件，从而产生了$-Δ_Ω^d $的非空离散频谱；特别是，我们表明，由于波导足够薄，离散特征值的数量可以任意大。
Let $Ω\subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-Δ_Ω^D$ the Dirichlet Laplacian operator in $Ω$. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-Δ_Ω^D$. Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for $-Δ_Ω^D$; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.
