{'fraction_non_alphanumeric': 0.02557200538358008, 'fraction_numerical': 0.0, 'mean_word_length': 5.148760330578512, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schrödinger Operators

Discrete Laplacians and discrete magnetic Schrödinger operators feature in many different areas of mathematics. They are used in combinatorics and computer science, appear as discretizations of (pseudo-)differential operators on Riemannian manifolds, serve as toy models for Hamiltonians in mathematical physics and play an important role in the study of random walks just to name a few. Even though discrete operators are used for very different means, their basic structure is always the same. Given a graph G = (X,E) consisting of vertices X and edges E one defines the discrete Laplacian ∆ acting on functions on X (up to a sign) by