On the Singular Behaviour of the Titchmarsh-Weyl m-Function for the Perturbed Hill’s Equation on the Line

For the perturbed Hill’s equation on the line, we study the behaviour of the matrix m-function at the spectral gap endpoints. In particular, we extend the result of Hinton, Klaus and Shaw that En , a gap endpoint, is a half-bound state (HBS) if and only if (E − En)½m(E) approaches a nonzero constant as E → En , to the present case.