In recent years, range-view-based LiDAR point cloud super-resolution techniques attract significant attention as a low-cost method for generating higher-resolution point cloud data. However, due to the sparsity and irregular structure of LiDAR point clouds, the point cloud super-resolution problem remains a challenging topic, especially for point cloud upsampling under novel views. In this paper, we propose SRMamba, a novel method for super-resolution of LiDAR point clouds in sparse scenes, addressing the key challenge of recovering the 3D spatial structure of point clouds from novel views. Specifically, we implement projection technique based on Hough Voting and Hole Compensation strategy to eliminate horizontally linear holes in range image. To improve the establishment of long-distance dependencies and to focus on potential geometric features in vertical 3D space, we employ Visual State Space model and Multi-Directional Scanning mechanism to mitigate the loss of 3D spatial structural information due to the range image. Additionally, an asymmetric U-Net network adapts to the input characteristics of LiDARs with different beam counts, enabling super-resolution reconstruction for multi-beam point clouds. We conduct a series of experiments on multiple challenging public LiDAR datasets (SemanticKITTI and nuScenes), and SRMamba demonstrates significant superiority over other algorithms in both qualitative and quantitative evaluations.