The spectrum of differential operators and square-integrable solutions

We give a comprehensive account of the relationship between the square-integrable solutions for real values of the spectral parameter λ and the spectrum of self-adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by Weidmann in his well-known 1987 monograph. According to a well-known result, if one endpoint is regular and for some real value of the spectral parameter λ the number of linearly independent square-integrable solutions is less than d, then λ is in the essential spectrum of every self-adjoint realization of the equation. Weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied. Here we prove this result without the additional condition.