Limit-point and limit-circle

Summary

For a singular Sturm–Liouville equation, each singular endpoint is either in the limit-point case or in the limit-circle case. In the limit-circle case every solution is square integrable near the endpoint; in the limit-point case some solution is not. This dichotomy, found by Hermann Weyl, tells whether a boundary condition is needed at that endpoint to get a self-adjoint operator. At a limit-point end no boundary condition is needed, and at a limit-circle end one must be imposed.