Graph Neural Networks (GNNs) excel in handling graph-structured data but often underperform in link prediction tasks compared to classical methods, mainly due to the limitations of the commonly used message-passing principle. Notably, their ability to distinguish non-isomorphic graphs is limited by the 1-dimensional Weisfeiler-Lehman test (WL). Our study presents a novel method to enhance the expressivity of GNNs by embedding induced subgraphs into the eigenbasis of the graph Laplacian. We introduce a Learnable Lanczos algorithm with Linear Constraints (LLwLC), proposing two novel subgraph extraction strategies: encoding vertex-deleted subgraphs and applying Neumann eigenvalue constraints. For the former, we demonstrate the ability to distinguish graphs that are indistinguishable by 2-WL, while maintaining efficiency. The latter focuses on link representations enabling differentiation between k-regular graphs and node automorphism, a vital aspect for link prediction tasks. Our approach results in a lightweight architecture, reducing the need for extensive training datasets. Empirically, our method improves performance in challenging link prediction tasks across benchmark datasets, establishing its practical utility and supporting our theoretical findings. Notably, LLwLC achieves 20x and 10x speedups by requiring only 5% and 10% of the data from the PubMed and OGBL-Vessel datasets, while comparing to the state-of-the-art.