A special class of solutions of the truncated Hill's equation

This work investigates the existence and properties of a certain class of solutions of the Hill's equation truncated in the interval [tau, tau + L] - where L = N a, a is the period of the coefficients in Hill's equation, N is a positive integer and tau is a real number. It is found that the truncated Hill's equation has two different types of solutions which vanish at the truncation boundaries tau and tau + L: There are always N-1 solutions in each stability interval of Hill's equation, whose eigen value is dependent on the truncation length L but not on the truncation boundary tau; There is always one and only one solution in each finite conditional instability interval of Hill's equation, whose eigen value might be dependent on the boundary location tau but not on the truncation length L. The results obtained are applied to the physics problem on the electronic states in one dimensional crystals of finite length. It significantly improves many known results and also provides more new understandings on the physics in low dimensional systems.