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The theory of electronic states in the traditionally solid state physics is essentially a theory of electronic states in crystals of infinite size. However, any real crystal always has a finite size. This book presents an analytical theory on the electronic states in ideal low-dimensional systems and finite crystals recently developed by the author based on a differential equation theory approach. It gives some exact and general fundamental understandings on the electronic states in ideal low-dimensional systems and finite crystals and provides new insights on some fundamental problems in low-dimensional systems such as the surface states, quantum confinement effects etc, some of them are quite different from what are traditionally believed in the solid state physics community.