Let \(\Omega \subset {\mathbb {R}}^3\) be a sheared waveguide, i.e., \(\Omega \) is built by translating a cross section in a constant direction along an unbounded spatial curve. Consider \(-\Delta _{\Omega }^D\) the Dirichlet Laplacian operator in \(\Omega \). Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of \(-\Delta _{\Omega }^D\). Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for \(-\Delta _{\Omega }^D\); in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.

令\（\ Omega \ subset {\ mathbb {R}} ^ 3 \）为剪切波导，即\（\ Omega \）是通过沿无界空间曲线沿恒定方向平移横截面而构建的。考虑\（-\ Delta _ {\ Omega} ^ D \）\（\ Omega \）中的Dirichlet拉普拉斯算子。在参考曲线的切线矢量在无穷远处具有有限极限的条件下，我们找到了\（-\ Delta _ {\ Omega} ^ D \）的基本谱。然后，我们陈述足够的条件，从而产生\（-\ Delta _ {\ Omega} ^ D \）的非空离散频谱；特别是，我们表明，由于波导足够细，离散特征值的数量可以任意大。