中文摘要：

目的研究H型群G上次Laplace算子的Dirichlet特征值问题。方法建立H型群G上向量场的性质，结合欧氏空间的经典方法。结果给出了H型群G上次Laplace算子Dirichlet特征值问题相邻特征值之差的估计，此结果与区域的几何和G的Lie代数的第二层的维数无关。结论把欧氏空间上的结论推广到了H型群上，并在H型群情形下有所深化。

英文摘要：

Aim To study the Dirichlet eigenvalue problem of the sub-Laplacian on groups of H type, G. Methods Establish properties of vector fields on groups of H type and combine the classical methods in the Euclidean spaces. Results The estimates for the differences of consecutive eigenvalues of the Dirichlet eigenvalue problem of the sub- Laplacian on groups of H type, G, are given. The results here are independent of the geometry of the domain and the dimension of the second layer of Lie algebra of G. Conclusion The conclusions in the Euclidean spaces are generalized to groups of H type and are deepen.